Search: a171250 -id:a171250
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A082381
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Sequence of the squared digital root of a number until 1 or 4 is reached. The initial numbers 1,2,..n are not output.
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+10
1
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1, 4, 9, 81, 65, 61, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 25, 29, 85, 89, 145, 42, 20, 4, 36, 45, 41, 17, 50, 25, 29, 85, 89, 145, 42, 20, 4, 49, 97, 130, 10, 1, 64, 52, 29, 85, 89, 145, 42, 20, 4, 81, 65, 61, 37, 58, 89, 145, 42, 20, 4, 1, 2, 4, 5, 25, 29
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OFFSET
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1,2
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COMMENTS
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Conjecture: The sequence always terminates with 1 or the 4 16 37 58 89 145 42 20 4... loop (cf. A080709).
This sequence should be read as fuzzy table, where the n-th row contains the successive results under the map "sum of digits squared", when starting with n, until either 1 or 4 is reached. So either of these two marks the end of a row: See example.
Row lengths (i.e. "stopping times") are given in A171250. (End)
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REFERENCES
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C. Stanley Ogilvy, Tomorrow's Math, 1972
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LINKS
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EXAMPLE
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The table reads:
[n=1] 1 (n=1 -> 1^2=1 -> STOP)
[n=2] 4 (n=2 -> 2^2=4 -> STOP)
[n=3] 9,81,65,61,37,58,89,145,42,20,4 (n=3 -> 3^2=9 -> 9^2=81 -> 8^2+1^2=65 -> ...)
[n=4] 16,37,58,89,145,42,20,4 (n=4 -> 4^2=16 -> 1^2+6^2=37 -> 3^2+7^2=58 -> ...)
...
[n=7] 49,97,130,10,1 (n=7 -> 7^2=49 -> 4^2+9^2=97 -> 130 -> 10 -> 1 -> STOP)
etc. (End)
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PROG
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(PARI) digitsq2(m) = {y=0; for(x=1, m, digitsq(x) ) }
/* The squared digital root of a number */ digitsq(n) = { while(1, s=0; while(n > 0, d=n%10; s = s+d*d; n=floor(n/10); ); print1(s" "); if(s==1 || s==4, break); n=s; ) }
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CROSSREFS
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Cf. A082382 (list also the initial value); sequences ending in the 4-loop: A000216 (n=2), A000218 (n=3), A080709 (n=4), A000221 (n=5), A008460 (n=6), A008462 (n=8), A008462 (n=9), A139566 (n=15), A122065 (n=74169); sequences ending in 1: A000012 (n=1), A008461 (n=7). [From M. F. Hasler, Dec 18 2009]
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KEYWORD
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nonn,base,easy,tabf
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AUTHOR
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EXTENSIONS
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Corrected and edited, added explanations M. F. Hasler, Dec 18 2009
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STATUS
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approved
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A178508
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a(n) = 1, 7, A011557*(period 6: repeat 10, 13, 31, 49, 70, 97).
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+10
1
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1, 7, 10, 13, 31, 49, 70, 97, 100, 130, 310, 490, 700, 970, 1000, 1300, 3100, 4900, 7000, 9700, 10000, 13000, 31000, 49000, 70000, 97000, 100000, 130000, 310000, 490000, 700000, 970000, 1000000, 1300000, 3100000, 4900000, 7000000, 9700000, 10000000
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f.: x*(10*x^2+13*x^3+31*x^4+49*x^5+60*x^6+27*x^7+1+7*x)/(1-10*x^6).
a(n) = 10^floor((n-3)/6)*(45+8*cos((n-2)*Pi)+25*cos((n-2)*Pi/3)+19*cos(2*(n-2)*Pi/3)-4*sqrt(3)*(4*sin((n-2)*Pi/3)+sin(2*(n-2)*Pi/3))) for n>2. - Wesley Ivan Hurt, Jun 18 2016
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MAPLE
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A178508:=n->10^floor((n-3)/6)*(45+8*cos((n-2)*Pi)+25*cos((n-2)*Pi/3)+19*cos(2*(n-2)*Pi/3)-4*sqrt(3)*(4*sin((n-2)*Pi/3)+sin(2*(n-2)*Pi/3))): 1, 7, seq(A178508(n), n=3..50); # Wesley Ivan Hurt, Jun 18 2016
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MATHEMATICA
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Join[{1, 7}, Flatten[Table[10^n {10, 13, 31, 49, 70, 97}, {n, 0, 6}]]] (* or *) Join[{1, 7}, LinearRecurrence[{0, 0, 0, 0, 0, 10}, {10, 13, 31, 49, 70, 97}, 50]] (* Harvey P. Dale, Nov 19 2013 *)
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PROG
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(PARI) a=[1, 7, 10, 13, 31, 49, 70, 97]; for(i=1, 99, a=concat(a, 10*a[#a-5])); a \\ Charles R Greathouse IV, Jun 01 2011
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Dissassociated with sums-of-squares sequences by R. J. Mathar, Jun 07 2010
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STATUS
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approved
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