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Search: a057958 -id:a057958
Displaying 11-20 of 21 results found. page 1 2 3
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A057952 Number of prime factors of 9^n - 1 (counted with multiplicity). +10
17
3, 5, 5, 7, 6, 8, 5, 10, 8, 10, 7, 11, 5, 9, 11, 12, 8, 12, 7, 13, 11, 11, 6, 17, 10, 9, 13, 13, 9, 17, 8, 14, 12, 12, 11, 16, 8, 11, 15, 18, 8, 18, 6, 16, 19, 10, 10, 21, 12, 18, 15, 13, 8, 18, 15, 19, 15, 13, 7, 24, 7, 13, 19, 16, 12, 18, 8, 17, 15, 20, 9, 24, 9, 13, 22, 17, 13, 22 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..690 (first 330 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project
FORMULA
Mobius transform of A085034. - T. D. Noe, Jun 19 2003
a(n) = A001222(A024101(n)) = A057958(2*n). - Amiram Eldar, Feb 02 2020
a(n) = A057941(n) + A057958(n). - Max Alekseyev, Jan 07 2024
MATHEMATICA
PrimeOmega[Table[9^n - 1, {n, 1, 30}]] (* Amiram Eldar, Feb 02 2020 *)
CROSSREFS
bigomega(b^n-1): A046051 (b=2), A057958 (b=3), A057957 (b=4), A057956 (b=5), A057955 (b=6), A057954 (b=7), A057953 (b=8), this sequence (b=9), A057951 (b=10), A366682 (b=11), A366708 (b=12).
Cf. A001222, A002591, A024101, A074477, A085034, A133801, A274909, A366660, A366661, A366662.
KEYWORD
nonn
AUTHOR
Patrick De Geest, Nov 15 2000
STATUS
approved
A057955 Number of prime factors of 6^n - 1 (counted with multiplicity). +10
17
1, 2, 2, 3, 3, 4, 2, 4, 4, 6, 3, 7, 3, 6, 6, 6, 5, 7, 3, 9, 4, 5, 5, 9, 6, 7, 6, 9, 2, 11, 3, 9, 6, 8, 7, 13, 6, 6, 6, 12, 3, 10, 5, 9, 11, 8, 4, 13, 5, 10, 9, 11, 4, 11, 7, 14, 7, 6, 4, 20, 4, 5, 10, 12, 9, 12, 3, 11, 8, 18, 2, 18, 5, 10, 12, 9, 6, 15, 4, 17, 8, 7, 8, 17, 10, 7, 7, 12, 4, 18, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..420
S. S. Wagstaff, Jr., The Cunningham Project
FORMULA
Möbius transform of A085031. - T. D. Noe, Jun 19 2003
a(n) = A001222(A024062(n)). - Amiram Eldar, Feb 02 2020
EXAMPLE
6^10 - 1 = 60466175 = 5^2 * 7 * 11 * 101 * 311 and a(10) = bigomega(60466175) = 2+1+1+1+1 = 6. - Bernard Schott, Feb 02 2020
MATHEMATICA
PrimeOmega[6^Range[100]-1] (* Harvey P. Dale, Dec 14 2015 *)
PROG
(Magma) f:=func<n|&+[p[2]: p in Factorization(n)]>; [f(6^n - 1):n in [1..90]]; // Marius A. Burtea, Feb 02 2020
CROSSREFS
bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), this sequence (b=6), A057956 (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).
Cf. A001222, A024062, A085031, A274907.
KEYWORD
nonn
AUTHOR
Patrick De Geest, Nov 15 2000
STATUS
approved
A057956 Number of prime factors of 5^n - 1 (counted with multiplicity). +10
17
2, 4, 3, 6, 4, 7, 3, 8, 5, 7, 3, 10, 3, 7, 7, 11, 4, 11, 5, 11, 6, 8, 4, 13, 8, 7, 9, 10, 5, 14, 4, 14, 6, 8, 9, 16, 5, 10, 6, 15, 4, 16, 4, 12, 12, 8, 3, 17, 4, 13, 8, 12, 5, 19, 10, 13, 7, 9, 4, 21, 5, 9, 11, 18, 8, 15, 7, 14, 9, 16, 4, 22, 5, 10, 16, 14, 7, 14, 5, 20, 11, 10, 5, 22, 9, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..502 (first 448 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project
FORMULA
Mobius transform of A085030. - T. D. Noe, Jun 19 2003
a(n) = A001222(A024049(n)). - Amiram Eldar, Feb 01 2020
MATHEMATICA
PrimeOmega[5^Range[90]-1] (* Harvey P. Dale, Dec 16 2017 *)
CROSSREFS
bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), this sequence (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).
Cf. A001222, A024049, A074479, A085030.
KEYWORD
nonn
AUTHOR
Patrick De Geest, Nov 15 2000
STATUS
approved
A085028 Number of prime factors of cyclotomic(n,3), which is A019321(n), the value of the n-th cyclotomic polynomial evaluated at x=3. +10
11
1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 2, 2, 2, 1, 2, 2, 1, 2, 1, 3, 2, 3, 2, 3, 2, 1, 3, 2, 1, 2, 2, 4, 1, 3, 3, 2, 2, 3, 1, 4, 3, 5, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 1, 2, 2, 1, 2, 3, 2, 3, 2, 2, 1, 1, 1, 4, 3, 3, 2, 3, 4, 3, 2, 3, 2, 4, 2, 2, 1, 3, 3, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The Mobius transform of this sequence yields A057958, number of prime factors of 3^n-1.
REFERENCES
See references at A085021
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..690
MATHEMATICA
Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 3]]][[2]], {n, 1, 100}]
CROSSREFS
omega(Phi(n,x)): A085021 (x=2), this sequence (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).
Cf. A019321, A057958, A059885, A143663, A218356, A074477.
KEYWORD
nonn
AUTHOR
T. D. Noe, Jun 19 2003
STATUS
approved
A173898 Decimal expansion of sum of the reciprocals of the Mersenne primes. +10
11
5, 1, 6, 4, 5, 4, 1, 7, 8, 9, 4, 0, 7, 8, 8, 5, 6, 5, 3, 3, 0, 4, 8, 7, 3, 4, 2, 9, 7, 1, 5, 2, 2, 8, 5, 8, 8, 1, 5, 9, 6, 8, 5, 5, 3, 4, 1, 5, 4, 1, 9, 7, 0, 1, 4, 4, 1, 9, 3, 1, 0, 6, 5, 2, 7, 3, 5, 6, 8, 7, 0, 1, 4, 4, 0, 2, 1, 2, 7, 2, 3, 4, 9, 9, 1, 5, 4, 8, 8, 3, 2, 9, 3, 6, 6, 6, 2, 1, 5, 3, 7, 4, 0, 3, 2, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
We know this a priori to be strictly less than the Erdős-Borwein constant (A065442), which Erdős (1948) showed to be irrational. This new constant would also seem to be irrational.
LINKS
Table of n, a(n) for n=0..105.
Peter B. Borwein, On the Irrationality of Certain Series, Math. Proc. Cambridge Philos. Soc. 112, 141-146, 1992.
Paul Erdős, On Arithmetical Properties of Lambert Series, J. Indian Math. Soc. 12, 63-66, 1948.
Yoshihiro Tanaka, On the Sum of Reciprocals of Mersenne Primes, American Journal of Computational Mathematics, Vol. 7, No. 2 (2017), pp. 145-148.
Eric Weisstein's World of Mathematics, Erdos-Borwein Constant.
Marek Wolf, Computer experiments with Mersenne primes, arXiv preprint arXiv:1112.2412 [math.NT], 2011.
FORMULA
Sum_{i>=1} 1/A000668(i).
EXAMPLE
Decimal expansion of (1/3) + (1/7) + (1/31) + (1/127) + (1/8191) + (1/131071) + (1/524287) + ... = .5164541789407885653304873429715228588159685534154197.
This has continued fraction expansion 0 + 1/(1 + 1/(1 + 1/(14 + 1/(1 + ...)))) (see A209601).
MAPLE
Digits := 120 ; L := [ 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917 ] ;
x := 0 ; for i from 1 to 30 do x := x+1.0/(2^op(i, L)-1 ); end do ;
MATHEMATICA
RealDigits[Sum[1/(2^p - 1), {p, MersennePrimeExponent[Range[14]]}], 10, 100][[1]] (* Amiram Eldar, May 24 2020 *)
PROG
(PARI) isM(p)=my(m=Mod(4, 2^p-1)); for(i=1, p-2, m=m^2-2); !m
s=1/3; forprime(p=3, default(realprecision)*log(10)\log(2), if(isM(p), s+=1./(2^p-1))); s \\ Charles R Greathouse IV, Mar 22 2012
CROSSREFS
Cf. A209601, A000668, A065442 (decimal expansion of Erdos-Borwein constant), A000043, A001348, A046051, A057951-A057958, A034876, A124477, A135659, A019279, A061652, A000225.
KEYWORD
cons,nonn
AUTHOR
Jonathan Vos Post, Mar 01 2010
EXTENSIONS
Entry revised by N. J. A. Sloane, Mar 10 2012
STATUS
approved
A366708 Number of prime factors of 12^n - 1 (counted with multiplicity). +10
9
1, 2, 2, 4, 2, 5, 3, 6, 4, 4, 4, 8, 3, 6, 6, 9, 3, 9, 2, 8, 5, 6, 4, 12, 4, 8, 6, 10, 5, 13, 5, 11, 8, 6, 9, 14, 3, 6, 7, 14, 4, 14, 5, 12, 12, 8, 3, 18, 5, 10, 6, 13, 7, 16, 8, 13, 7, 8, 4, 19, 4, 8, 8, 13, 8, 17, 5, 10, 7, 14, 4, 21, 3, 7, 11, 11, 11, 18, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..310
FORMULA
a(n) = bigomega(12^n-1) = A001222(A024140(n)).
MATHEMATICA
PrimeOmega[12^Range[70]-1]
PROG
(PARI) a(n)=bigomega(12^n-1)
CROSSREFS
Cf. A024140, A001222, A046051, A057951, A057952, A057953, A057954, A057955, A057956, A057957, A057958, A250288, A252170, A366707, A366709, A366682.
KEYWORD
nonn
AUTHOR
Sean A. Irvine, Oct 17 2023
STATUS
approved
A366682 Number of prime factors of 11^n - 1 (counted with multiplicity). +10
7
2, 5, 4, 7, 4, 9, 4, 9, 5, 8, 4, 13, 4, 8, 7, 12, 3, 12, 3, 11, 10, 11, 5, 17, 8, 10, 6, 13, 4, 15, 5, 15, 9, 9, 8, 17, 6, 10, 12, 15, 9, 17, 4, 15, 9, 12, 5, 24, 7, 14, 9, 13, 6, 16, 10, 19, 8, 10, 5, 21, 5, 12, 16, 19, 8, 22, 6, 15, 10, 19, 7, 24, 3, 11, 15 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..316
FORMULA
a(n) = bigomega(11^n-1) = A001222(A024127(n)).
MATHEMATICA
PrimeOmega[11^Range[70]-1]
PROG
(PARI) a(n)=bigomega(11^n-1)
CROSSREFS
Cf. A024127, A001222, A366681, A366683, A366684, A366685, A366687.
Cf. A046051, A057958, A057957, A057956, A057955, A057954, A057953, A057952, A057951, A366708.
KEYWORD
nonn
AUTHOR
Sean A. Irvine, Oct 16 2023
STATUS
approved
A143663 a(n) is the least prime such that the multiplicative order of 3 mod a(n) equals n, or a(n)=1 if no such prime exists. +10
6
2, 1, 13, 5, 11, 7, 1093, 41, 757, 61, 23, 73, 797161, 547, 4561, 17, 1871, 19, 1597, 1181, 368089, 67, 47, 6481, 8951, 398581, 109, 29, 59, 31, 683, 21523361, 2413941289, 103, 71, 530713, 13097927, 2851, 313, 42521761, 83, 43, 431, 5501, 181, 23535794707 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
If a(n) differs from 1, then a(n) is the minimal prime divisor of A064079(n).
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..730 (first 153 terms from Robert G. Wilson v)
MAPLE
a:= proc(n) local f, p;
f:= numtheory:-factorset(3^n - 1);
for p in f do
if numtheory:-order(3, p) = n then return p fi
od:
1
end proc:
seq(a(n), n=1..100); # Robert Israel, Oct 13 2014
MATHEMATICA
p = 2; t = Table[0, {100}]; While[p < 100000001, a = MultiplicativeOrder[3, p]; If[0 < a < 101 && t[[a]] == 0, t[[a]] = p; Print[{a, p}]]; p = NextPrime@ p]; t (* Robert G. Wilson v, Oct 13 2014 *)
CROSSREFS
Cf. A002326, A064079, A112927, A112092, A143665, A007138, A218356, A057958, A074477.
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Aug 28 2008
EXTENSIONS
More terms from Robert G. Wilson v, Dec 11 2013
STATUS
approved
A002591 Largest prime factor of 3^(2n+1) - 1.
(Formerly M4886 N2095) 		+10
3
2, 13, 11, 1093, 757, 3851, 797161, 4561, 34511, 363889, 368089, 1001523179, 391151, 8209, 20381027, 4404047, 2413941289, 2664097031, 17189128703, 797161, 86950696619, 380808546861411923, 927001, 96656723, 131713, 99810171997 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
REFERENCES
J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 2, p. 28.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Table of n, a(n) for n = 0..344
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
S. S. Wagstaff, Jr., The Cunningham Project
FORMULA
a(n) = A074477(2n+1). - Max Alekseyev, May 22 2022
MATHEMATICA
Table[FactorInteger[3^(2n-1)-1][[-1, 1]], {n, 30}] (* Harvey P. Dale, Oct 19 2022 *)
CROSSREFS
Together with A274909 forms bisection of A074477.Cf. A057958, A059885, A085028, A133801, A235366.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane
EXTENSIONS
Corrected and extended by Jud McCranie, Jan 03 2001
Terms up to a(307) in b-file from Sean A. Irvine, Apr 20 2014
a(0) prepended and a(308)-a(344) added to b-file by Max Alekseyev, Apr 24 2019, Sep 10 2020, Aug 26 2021, May 22 2022
STATUS
approved
A109472 Cumulative sum of primes p such that 2^p - 1 is a Mersenne prime. +10
2
2, 5, 10, 17, 30, 47, 66, 97, 158, 247, 354, 481, 1002, 1609, 2888, 5091, 7372, 10589, 14842, 19265, 28954, 38895, 50108, 70045, 91746, 114955, 159452, 245695, 356198, 488247, 704338, 1461177, 2320610, 3578397, 4976666, 7952887, 10974264, 17946857, 31413774, 52409785, 76446368, 102411319, 132813776, 165396433, 202553100, 245196901, 288309510 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Prime cumulative sum of primes p such that 2^p - 1 is a Mersenne prime include: a(1) = 2, a(2) = 5, a(4) = 17, a(6) = 47, a(8) = 97, a(14) = 1609, a(18) = 10589. After 1, all such indices x of prime a(x) must be even.
LINKS
Gord Palameta, Table of n, a(n) for n = 1..47
FORMULA
a(n) = Sum_{i=1..n} A000043(i).
EXAMPLE
a(1) = 2, since 2^2-1 = 3 is a Mersenne prime.
a(2) = 2 + 3 = 5, since 2^3-1 = 7 is a Mersenne prime.
a(3) = 2 + 3 + 5 = 10, since 2^5-1 = 31 is a Mersenne prime.
a(4) = 2 + 3 + 5 + 7 = 17, since 2^7-1 = 127 is a Mersenne prime; 17 itself is prime (in fact a p such that 2^p-1 is a Mersenne prime).
a(18) = 2 + 3 + 5 + 7 + 13 + 17 + 19 + 31 + 61 + 89 + 107 + 127 + 521 + 607 + 1279 + 2203 + 2281 + 3217 = 10589 (which is prime).
MATHEMATICA
Accumulate[Select[Range[3000], PrimeQ[2^# - 1] &]] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)
Accumulate@ MersennePrimeExponent@ Range@ 45 (* Michael De Vlieger, Jul 22 2018 *)
CROSSREFS
Cf. A000043, A000668 for the Mersenne primes, A001348, A046051, A057951-A057958.
KEYWORD
nonn
AUTHOR
Jonathan Vos Post, Aug 28 2005
EXTENSIONS
a(38)-a(47) from Gord Palameta, Jul 21 2018
STATUS
approved
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Last modified August 29 23:09 EDT 2024. Contains 375519 sequences. (Running on oeis4.)