Search: a133619 -id:a133619
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A133612
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Unique sequence of digits a(0), a(1), a(2), ... such that for all k >= 2, the number A(k) := Sum_{n = 0..k-1} a(n)*10^n satisfies 2^A(k) == A(k) (mod 10^k).
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+10
25
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6, 3, 7, 8, 4, 9, 2, 3, 4, 3, 5, 3, 5, 7, 0, 5, 1, 6, 8, 9, 0, 8, 3, 3, 3, 5, 8, 9, 5, 1, 0, 0, 6, 2, 7, 8, 6, 9, 6, 8, 2, 5, 5, 4, 1, 0, 7, 5, 4, 2, 6, 8, 2, 6, 1, 4, 8, 2, 8, 2, 1, 2, 1, 2, 1, 9, 0, 7, 2, 9, 8, 3, 5, 5, 8, 9, 8, 9, 7, 1, 0, 4, 9, 0, 5, 2, 2, 0, 9, 1, 7, 8, 8, 8, 6, 5, 2, 2, 4, 4, 8, 3, 7, 1, 0
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OFFSET
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0,1
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COMMENTS
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10-adic expansion of the iterated exponential 2^^n (A014221) for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n > 9, 2^^n == 2948736 (mod 10^7).
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REFERENCES
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M. Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6
Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.
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LINKS
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EXAMPLE
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63784923435357051689083335895100627869682554107542682614828212121907298... - Robert G. Wilson v, Feb 22 2014
2^36 = 68719476736 == 36 (mod 100), 2^736 == 736 (mod 1000), 2^8736 == 8736 (mod 10000), etc.
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MATHEMATICA
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(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[2, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Feb 22 2014 *)
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CROSSREFS
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Cf. A133613, A133614, A133615, A133616, A133617, A133618, A133619, A144539, A144540, A144541, A144542, A144543, A144544.
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KEYWORD
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nonn,base
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AUTHOR
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Daniel Geisler (daniel(AT)danielgeisler.com), Dec 18 2007
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EXTENSIONS
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More terms from J. Luis A. Yebra, Dec 12 2008
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STATUS
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approved
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A133613
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Decimal digits such that for all k >= 1, the number A(k) := Sum_{n = 0..k-1} a(n)*10^n satisfies the congruence 3^A(k) == A(k) (mod 10^k).
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+10
18
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7, 8, 3, 5, 9, 1, 4, 6, 4, 2, 6, 2, 7, 2, 6, 5, 7, 5, 4, 0, 1, 9, 5, 0, 9, 3, 4, 6, 8, 1, 5, 8, 4, 8, 1, 0, 7, 6, 9, 3, 2, 7, 8, 4, 3, 2, 2, 2, 3, 0, 0, 8, 3, 6, 6, 9, 4, 5, 0, 9, 7, 6, 9, 3, 9, 9, 8, 1, 6, 9, 9, 3, 6, 9, 7, 5, 3, 5, 2, 6, 5, 1, 5, 8, 3, 9, 1, 8, 1, 0, 5, 6, 2, 8, 4, 2, 4, 0, 4, 9, 8, 0, 5, 1, 6
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OFFSET
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0,1
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COMMENTS
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10-adic expansion of the iterated exponential 3^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n>9, 3^^n == 4195387 (mod 10^7).
This sequence also gives many final digits of Graham's number ...399618993967905496638003222348723967018485186439059104575627262464195387. - Paul Muljadi, Sep 08 2008 and J. Luis A. Yebra, Dec 22 2008
Graham's number can be represented as G(64):=3^^3^^...^^3 [see M. Gardner and Wikipedia], in which case its G(63) lowermost digits are guaranteed to match this sequence (i.e., the convergence speed of the base 3 is unitary - see A317905). To avoid such confusion, it would be best to interpret this sequence as a real-valued constant 0.783591464..., corresponding to 3^^k in the limit of k->infinity, and call it Graham's constant G(3). Generalizations to G(n) and G(n,base) are obvious. - Stanislav Sykora, Nov 07 2015
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REFERENCES
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M. Gardner, Mathematical Games, Scientific American 237, 18 - 28 (1977).
M. Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, p. 11-12, 69-78. ISBN 978-88-6178-789-6.
Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.
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LINKS
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FORMULA
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a(n) = floor( A183613(n+1) / 10^n ).
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EXAMPLE
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783591464262726575401950934681584810769327843222300836694509769399816993697535...
Consider the sequence 3^^n: 1, 3, 27, 7625597484987, ... From 3^^3 = 7625597484987 onwards, all terms end with the digits 87. This follows from Euler's generalization of Fermat's little theorem.
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MATHEMATICA
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(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[3, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)
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CROSSREFS
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Cf. A133612, A133614, A133615, A133616, A133617, A133618, A133619, A144539, A144540, A144541, A144542, A144543, A144544, A317905, A318478.
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KEYWORD
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nonn,base
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AUTHOR
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Daniel Geisler (daniel(AT)danielgeisler.com), Dec 18 2007
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EXTENSIONS
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More terms from J. Luis A. Yebra, Dec 12 2008
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STATUS
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approved
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A133614
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Unique sequence of digits a(0), a(1), a(2), ... such that for all k >= 2, the number A(k) := Sum_{n = 0..k-1} a(n)*10^n satisfies 4^A(k) == A(k) (mod 10^k).
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+10
17
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6, 9, 8, 8, 2, 7, 1, 1, 4, 0, 9, 2, 5, 5, 5, 2, 0, 3, 2, 2, 6, 3, 9, 4, 9, 5, 3, 1, 4, 3, 9, 3, 1, 2, 0, 6, 5, 7, 5, 6, 3, 4, 2, 1, 3, 5, 2, 6, 0, 6, 2, 9, 5, 4, 0, 6, 6, 0, 7, 5, 9, 5, 6, 9, 0, 6, 1, 4, 6, 8, 8, 3, 8, 3, 6, 4, 8, 8, 0, 5, 2, 3, 0, 3, 2, 6, 2, 5, 4, 1, 1, 1, 9, 0, 9, 8, 0, 8, 1, 4, 3, 1, 0, 1, 8
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OFFSET
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0,1
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COMMENTS
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10-adic expansion of the iterated exponential 4^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n > 9, 4^^n == 1728896 (mod 10^7).
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REFERENCES
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M. Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, p. 69-78. ISBN 978-88-6178-789-6.
Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.
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LINKS
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EXAMPLE
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698827114092555203226394953143931206575634213526062954066075956906146883836488...
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MATHEMATICA
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(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[4, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)
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CROSSREFS
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Cf. A133612, A133613, A133615, A133616, A133617, A133618, A133619, A144539, A144540, A144541, A144542, A144543, A144544.
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KEYWORD
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nonn,base
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AUTHOR
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Daniel Geisler (daniel(AT)danielgeisler.com), Dec 18 2007
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EXTENSIONS
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More terms from J. Luis A. Yebra, Dec 12 2008
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STATUS
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approved
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A133615
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Unique sequence of digits a(0), a(1), a(2), ... such that for all k >= 2, the number A(k) := Sum_{n = 0..k-1} a(n)*10^n satisfies 5^A(k) == A(k) (mod 10^k).
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+10
17
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5, 2, 1, 3, 0, 2, 8, 0, 4, 8, 1, 6, 2, 5, 1, 3, 9, 4, 7, 1, 1, 7, 8, 5, 3, 8, 0, 9, 5, 1, 1, 5, 6, 9, 8, 0, 4, 9, 2, 2, 9, 8, 9, 3, 3, 9, 8, 1, 3, 3, 1, 7, 7, 4, 6, 7, 1, 0, 2, 8, 3, 7, 5, 1, 7, 3, 1, 4, 1, 1, 9, 7, 8, 2, 9, 6, 2, 5, 5, 5, 3, 3, 0, 9, 0, 4, 7, 3, 1, 8, 5, 7, 4, 6, 9, 7, 2, 3, 0, 8, 9, 2, 6, 1, 4
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graph;
refs;
listen;
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text;
internal format)
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OFFSET
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0,1
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COMMENTS
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10-adic expansion of the iterated exponential 5^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n > 9, 5^^n == 8203125 (mod 10^7).
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REFERENCES
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M. Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, p. 69-78. ISBN 978-88-6178-789-6.
Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.
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LINKS
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EXAMPLE
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521302804816251394711785380951156980492298933981331774671028375173141197829625...
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MATHEMATICA
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(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[5, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)
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CROSSREFS
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Cf. A133612, A133613, A133614, A133616, A133617, A133618, A133619, A144539, A144540, A144541, A144542, A144543, A144544.
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KEYWORD
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nonn,base
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AUTHOR
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Daniel Geisler (daniel(AT)danielgeisler.com), Dec 18 2007
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EXTENSIONS
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More terms from J. Luis A. Yebra, Dec 12 2008
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STATUS
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approved
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A133617
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Unique sequence of digits a(0), a(1), a(2), ... such that for all k >= 2, the number A(k) := Sum_{n = 0..k-1} a(n)*10^n satisfies 7^A(k) == A(k) (mod 10^k).
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+10
17
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3, 4, 3, 2, 7, 1, 5, 6, 5, 1, 1, 5, 5, 6, 2, 1, 3, 3, 3, 4, 6, 3, 5, 8, 3, 3, 3, 7, 3, 6, 0, 8, 6, 0, 3, 6, 9, 5, 6, 7, 4, 1, 8, 2, 6, 6, 5, 9, 2, 6, 5, 3, 0, 8, 6, 5, 2, 8, 4, 4, 4, 7, 7, 7, 6, 7, 5, 4, 9, 1, 2, 9, 8, 6, 5, 7, 7, 0, 7, 8, 4, 2, 6, 3, 8, 5, 4, 8, 1, 9, 4, 5, 8, 3, 9, 9, 5, 4, 4, 0, 3, 8, 2, 2, 0
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listen;
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text;
internal format)
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OFFSET
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0,1
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COMMENTS
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10-adic expansion of the iterated exponential 7^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n > 9, 7^^n == 5172343 (mod 10^7).
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REFERENCES
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M. Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, p. 69-78. ISBN 978-88-6178-789-6.
Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.
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LINKS
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EXAMPLE
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343271565115562133346358333736086036956741826659265308652844477767549129865770...
Sequences A133612-A144544 generalize the observation that 7^343 == 343 (mod 1000).
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MATHEMATICA
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(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[7, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)
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CROSSREFS
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Cf. A133612, A133613, A133614, A133615, A133616, A133618, A133619, A144539, A144540, A144541, A144542, A144543, A144544.
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KEYWORD
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nonn,base
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AUTHOR
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Daniel Geisler (daniel(AT)danielgeisler.com), Dec 18 2007
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EXTENSIONS
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More terms from J. Luis A. Yebra, Dec 12 2008
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STATUS
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approved
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A133618
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Unique sequence of digits a(0), a(1), a(2), ... such that for all k >= 2, the number A(k) := Sum_{n = 0..k-1} a(n)*10^n satisfies 8^A(k) == A(k) (mod 10^k).
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+10
17
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6, 5, 8, 5, 2, 2, 5, 9, 8, 6, 1, 4, 5, 3, 0, 7, 7, 5, 1, 2, 5, 1, 8, 0, 0, 1, 5, 8, 8, 5, 5, 9, 0, 2, 6, 1, 3, 9, 1, 1, 5, 6, 2, 9, 8, 3, 7, 7, 2, 0, 1, 5, 7, 3, 8, 8, 2, 6, 6, 7, 0, 3, 7, 5, 7, 2, 7, 4, 2, 4, 4, 2, 4, 3, 7, 5, 8, 4, 4, 2, 2, 1, 3, 0, 8, 8, 8, 8, 7, 1, 5, 9, 1, 2, 0, 1, 6, 0, 9, 8, 0, 5, 3, 3, 1
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graph;
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listen;
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text;
internal format)
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OFFSET
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0,1
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COMMENTS
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10-adic expansion of the iterated exponential 8^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n > 9, 8^^n == 5225856 (mod 10^7).
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REFERENCES
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M. Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, p. 69-78. ISBN 978-88-6178-789-6.
Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.
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LINKS
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EXAMPLE
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658522598614530775125180015885590261391156298377201573882667037572742442437584...
8^56 == 56 (mod 100), 8^856 == 856 (mod 1000), ...
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MATHEMATICA
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(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[8, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)
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CROSSREFS
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Cf. A133612, A133613, A133614, A133615, A133616, A133617, A133619, A144539, A144540, A144541, A144542, A144543, A144544.
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KEYWORD
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nonn,base
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AUTHOR
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Daniel Geisler (daniel(AT)danielgeisler.com), Dec 18 2007
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EXTENSIONS
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More terms from J. Luis A. Yebra, Dec 12 2008
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STATUS
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approved
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A133616
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Unique sequence of digits a(0), a(1), a(2), ... such that for all k >= 2, the number A(k) := Sum_{n = 0..k-1} a(n)*10^n satisfies 6^A(k) == A(k) (mod 10^k).
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+10
16
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6, 5, 6, 8, 3, 2, 7, 4, 4, 7, 2, 2, 3, 9, 5, 5, 6, 9, 7, 6, 7, 3, 2, 1, 9, 6, 0, 1, 7, 5, 0, 6, 0, 5, 8, 6, 9, 1, 8, 0, 1, 3, 7, 9, 4, 6, 0, 4, 4, 7, 0, 4, 6, 4, 0, 2, 4, 6, 3, 7, 8, 1, 6, 7, 0, 8, 5, 0, 1, 4, 3, 4, 4, 4, 1, 8, 5, 7, 5, 9, 7, 0, 0, 4, 2, 9, 6, 3, 4, 1, 8, 9, 6, 0, 9, 8, 4, 5, 7, 0, 3, 5, 0, 8, 6
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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COMMENTS
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10-adic expansion of the iterated exponential 6^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n > 9, 6^^n == 7238656 (mod 10^7).
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REFERENCES
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M. Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, p. 69-78. ISBN 978-88-6178-789-6.
Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.
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LINKS
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EXAMPLE
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656832744722395569767321960175060586918013794604470464024637816708501434441857...
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MATHEMATICA
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(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[6, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)
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CROSSREFS
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Cf. A133612, A133613, A133614, A133615, A133617, A133618, A133619, A144539, A144540, A144541, A144542, A144543, A144544.
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KEYWORD
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nonn,base
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AUTHOR
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Daniel Geisler (daniel(AT)danielgeisler.com), Dec 18 2007
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EXTENSIONS
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More terms from J. Luis A. Yebra, Dec 12 2008
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STATUS
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approved
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A144539
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Unique sequence of digits a(0), a(1), a(2), .. such that for all k >= 2, the number A(k) := Sum_{n = 0..k-1 } a(n)*10^n satisfies 11^A(k) == A(k) mod 10^k.
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+10
16
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1, 1, 6, 6, 6, 6, 2, 7, 1, 9, 7, 8, 3, 0, 7, 6, 6, 2, 0, 2, 7, 1, 9, 8, 7, 9, 3, 4, 3, 2, 6, 9, 8, 1, 1, 7, 5, 1, 0, 2, 0, 4, 5, 9, 4, 3, 9, 9, 9, 4, 5, 3, 9, 3, 9, 2, 4, 3, 8, 4, 1, 6, 0, 5, 6, 8, 8, 0, 6, 4, 2, 9, 2, 6, 1, 6, 6, 4, 0, 9, 0, 3, 9, 4, 9, 6, 8, 9, 0, 8, 6, 9, 1, 8, 7, 5, 0, 5, 8, 6, 7, 4, 6, 5, 3
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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REFERENCES
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M. Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, p. 69-78. ISBN 978-88-6178-789-6.
Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.
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LINKS
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EXAMPLE
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116666271978307662027198793432698117510204594399945393924384160568806429261664...
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MATHEMATICA
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(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[11, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)
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CROSSREFS
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Cf. A133612, A133613, A133614, A133615, A133616, A133617, A133618, A133619, A144540, A144541, A144542, A144543, A144544.
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A144540
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Unique sequence of digits a(0), a(1), a(2), .. such that for all k >= 2, the number A(k) := Sum_{n = 0..k-1 } a(n)*10^n satisfies 12^A(k) == A(k) mod 10^k.
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+10
16
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6, 1, 4, 2, 1, 0, 4, 5, 4, 1, 2, 4, 4, 1, 7, 1, 3, 9, 4, 8, 5, 8, 4, 8, 5, 3, 1, 9, 5, 3, 6, 9, 3, 2, 5, 7, 1, 9, 7, 7, 7, 8, 2, 3, 3, 9, 4, 2, 1, 0, 4, 8, 5, 7, 9, 6, 7, 9, 5, 6, 2, 5, 3, 5, 7, 6, 7, 5, 4, 1, 5, 0, 8, 2, 9, 0, 3, 6, 4, 1, 0, 7, 9, 6, 2, 8, 3, 8, 0, 3, 4, 4, 1, 4, 6, 4, 3, 9, 0, 4, 2, 3, 0, 5, 9
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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REFERENCES
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M. Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, p. 69-78. ISBN 978-88-6178-789-6.
Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.
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LINKS
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EXAMPLE
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614210454124417139485848531953693257197778233942104857967956253576754150829036...
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MATHEMATICA
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(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[12, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)
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CROSSREFS
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Cf. A133612, A133613, A133614, A133615, A133616, A133617, A133618, A133619, A144539, A144541, A144542, A144543, A144544.
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A144541
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Unique sequence of digits a(0), a(1), a(2), .. such that for all k >= 2, the number A(k) := Sum_{n = 0..k-1 } a(n)*10^n satisfies 13^A(k) == A(k) mod 10^k.
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+10
16
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3, 5, 0, 5, 4, 0, 5, 5, 2, 8, 8, 4, 5, 9, 1, 8, 1, 2, 2, 4, 8, 8, 8, 7, 6, 9, 2, 0, 7, 1, 6, 8, 6, 9, 0, 4, 6, 7, 3, 2, 3, 5, 6, 8, 9, 4, 4, 3, 6, 6, 5, 6, 6, 3, 5, 9, 3, 1, 7, 0, 4, 3, 3, 7, 4, 6, 1, 4, 7, 6, 6, 9, 7, 2, 8, 5, 4, 7, 9, 3, 5, 5, 6, 7, 6, 5, 5, 8, 2, 1, 5, 0, 2, 2, 5, 4, 0, 5, 6, 8, 2, 7, 1, 8, 6
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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REFERENCES
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M. Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, p. 69-78. ISBN 978-88-6178-789-6.
Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.
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LINKS
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EXAMPLE
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350540552884591812248887692071686904673235689443665663593170433746147669728547...
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MATHEMATICA
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(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[13, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)
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CROSSREFS
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Cf. A133612, A133613, A133614, A133615, A133616, A133617, A133618, A133619, A144539, A144540, A144542, A144543, A144544.
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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