A065341 -id:A065341 - cp's OEIS frontend

A135975 Number of prime factors (without multiplicity) in Mersenne composites A065341.

2, 2, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 5, 4, 5, 2, 4, 3, 4, 5, 3, 2, 2, 3, 6, 2, 4, 4, 6, 2, 5, 3, 4, 2, 2, 3, 2, 3, 2, 5, 3, 4, 4, 3, 5, 2, 3, 3, 6, 5, 2, 2, 5, 3, 9, 4, 3, 5, 2, 8, 4, 4, 3, 5, 2, 4, 6, 3, 4, 2, 7, 3, 4, 4, 2, 5, 4, 5, 3, 5, 4, 3, 6, 4, 3, 4, 3, 4, 4

Offset: 1

Artur Jasinski, Dec 09 2007

nonn

Currently the smallest prime exponent p for which 2^p-1 is incompletely factored is p = 1213. - Gord Palameta, Aug 06 2018

Gord Palameta, Table of n, a(n) for n = 1..183
GIMPS, Status of M1213
S. S. Wagstaff, Jr., Main Tables from the Cunningham Project: cofactor of M1213 is C297.

Cf. A000225, A001221, A065341, A054723, A134852.

k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; AppendTo[k, d]], {n, 1, 40}]; k
(PrimeNu /@ Select[2^Prime[Range[40]] - 1, ! PrimeQ[#] &]) (* Jean-François Alcover, Aug 13 2014 *)

forprime(p=1, 1e3, if(!ispseudoprime(2^p-1), print1(omega(2^p-1), ", "))) \\ Felix Fröhlich, Aug 12 2014

a(n) = A001221(A065341(n)). - Michel Marcus, Aug 07 2018

a(29)-a(46) from Felix Fröhlich, Aug 12 2014

a(47)-a(100) from Gord Palameta, Aug 07 2018

A135977 Mersenne composites (A065341) with exactly 3 prime factors.

Original entry on oeis.org

536870911, 8796093022207, 140737488355327, 9007199254740991, 2361183241434822606847, 9444732965739290427391, 604462909807314587353087

Offset: 1

Artur Jasinski, Dec 09 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..42

Subsequence of A065341.

Cf. A000225, A054723, A134852, A135975, A135976, A135978, A135979, A344515.

k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d == 3, AppendTo[k, 2^Prime[n] - 1]]], {n, 1, 40}]; k

a(n) = 2^A344515(n) - 1. - Amiram Eldar, May 23 2021

A135976 Mersenne composites (A065341) with exactly 2 prime factors.

Original entry on oeis.org

2047, 8388607, 137438953471, 2199023255551, 576460752303423487, 147573952589676412927, 9671406556917033397649407, 158456325028528675187087900671, 2535301200456458802993406410751

Offset: 1

Artur Jasinski, Dec 09 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..39
Wikipedia, Semiprime.

Subsequence of A065341.

Cf. A000225, A054723, A134852, A135975, A135977, A135978, A135979.

A135976 := proc(n) local i;
i := 2^(ithprime(n))-1:
if (nops(numtheory[factorset](i)) = 2) then
   RETURN (i)
fi: end: [ seq(A135976(n), n=1..26) ]; # Jani Melik, Feb 09 2011

k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d == 2, AppendTo[k, 2^Prime[n] - 1]]], {n, 1, 40}]; k

forprime(p=1, 1e2, if(bigomega(2^p-1)==2, print1(2^p-1, ", "))) \\ Felix Fröhlich, Aug 12 2014

a(n) = 2^A135978(n) - 1. - Amiram Eldar, May 23 2021

A145097 a(n) is the largest proper divisor of the Mersenne composite A065341(n).

Original entry on oeis.org

89, 178481, 2304167, 616318177, 164511353, 20408568497, 59862819377, 1416003655831, 3203431780337, 761838257287, 10334355636337793, 21514198099633918969, 224958284260258499201, 57912614113275649087721

Offset: 1

Artur Jasinski, Oct 01 2008

nonn

Note that not all the largest divisors are primes.

Which divisors are prime? - see A145099. - Artur Jasinski, Oct 04 2008

Amiram Eldar, Table of n, a(n) for n = 1..183

Cf. A065341, A135975, A135980, A136031.

a = {}; Do[m = 2^Prime[n] - 1; If[PrimeQ[m], null, AppendTo[a, Divisors[m][[ -2]]]], {n, 1, 40}]; a

Name clarified by Amiram Eldar, Mar 12 2020

A136032 Number of prime factors (with multiplicity) of Mersenne composites (A065341).

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 5, 4, 5, 2, 4, 3, 4, 5, 3, 2, 2, 3, 6, 2, 4, 4, 6, 2, 5, 3, 4, 2, 2, 3, 2, 3, 2, 5, 3, 4, 4, 3, 5, 2, 3, 3, 6, 5, 2, 2, 5, 3, 9, 4, 3, 5, 2, 8, 4, 4, 3, 5, 2, 4, 6, 3, 4, 2, 7, 3, 4, 4, 2, 5, 4, 5, 3, 5, 4

Offset: 1

Artur Jasinski, Dec 11 2007

nonn

If the conjecture that all Mersenne composites are squarefree is true, then this sequence is identical to A135975. - Felix Fröhlich, Aug 24 2014

Amiram Eldar, Table of n, a(n) for n = 1..183

Cf. A001222, A065341, A135975.

a = {}; Do[If[PrimeQ[n] && !PrimeQ[2^n - 1], w = 2^n - 1; c = FactorInteger[w]; d = Length[c]; b = 0; Do[b = b + c[[k]][[2]], {k, 1, d}]; AppendTo[a, b]], {n, 2, 150}]; a
PrimeOmega/@Select[2^Prime[Range[100]]-1,!PrimeQ[#]&] (* Harvey P. Dale, Nov 01 2016 *)

forprime(p=2, 1e3, if(!ispseudoprime(2^p-1), print1(bigomega(2^p-1), ", "))) \\ Felix Fröhlich, Aug 24 2014

a(n) = A001222(A065341(n)). - Michel Marcus, Aug 24 2014

More terms from Michel Marcus, Nov 04 2013
Definition adjusted by Felix Fröhlich, Aug 24 2014
More terms from Felix Fröhlich, Aug 24 2014

A135386 Mersenne composites A065341 with 4 or more prime factors.

Original entry on oeis.org

10384593717069655257060992658440191, 2854495385411919762116571938898990272765493247, 182687704666362864775460604089535377456991567871

Offset: 1

Artur Jasinski, Dec 09 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..102

Subsequence of A065341.

Cf. A000225, A054723, A134852, A135975, A135976, A135977, A135979.

A135386 := proc(n) local i;
i := 2^(ithprime(n))-1:
if (nops(numtheory[factorset](i)) > 3) then
   RETURN (i)
fi: end: seq(A135386(n), n=1..37); # Jani Melik, Feb 09 2011

k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d >3, AppendTo[k, 2^Prime[n] - 1]]], {n, 1, 40}]; k

A001348 Mersenne numbers: 2^p - 1, where p is prime.

Original entry on oeis.org

3, 7, 31, 127, 2047, 8191, 131071, 524287, 8388607, 536870911, 2147483647, 137438953471, 2199023255551, 8796093022207, 140737488355327, 9007199254740991, 576460752303423487, 2305843009213693951, 147573952589676412927, 2361183241434822606847

Offset: 1

N. J. A. Sloane

Mersenne numbers A000225 whose indices are primes. - Omar E. Pol, Aug 31 2008
All terms are of the form 4k-1. - Paul Muljadi, Jan 31 2011
Smallest number with Hamming weight A000120 = prime(n). - M. F. Hasler, Oct 16 2018
The 5th, 9th, 10th, ... terms are not prime. See A000668 and A065341 for the primes and for the composites in this sequence. - M. F. Hasler, Nov 14 2018 [corrected by Jerzy R Borysowicz, Apr 08 2025]
Except for the first term 3: all prime factors of 2^p-1 must be 1 or -1 (mod 8), and 1 (mod 2p). - William Hu, Mar 10 2024

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 16.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 47.
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).

T. D. Noe, Table of n, a(n) for n = 1..100
Raymond Clare Archibald, Mersenne's Numbers, Scripta Mathematica, Vol. 3 (1935), pp. 112-119.
John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman and S. S. Wagstaff, Jr., Cunningham Project [Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers]
C. K. Caldwell, Mersenne Primes
Will Edgington, Mersenne Page> [from Internet Archive Wayback Machine].
Graham Everest, Shaun Stevens, Duncan Tamsett and Tom Ward, Primes generated by recurrence sequences, Amer. Math. Monthly, Vol. 114, No. 5 (2007), pp. 417-431.
Paul Garrett, Lucas-Lehmer criterion for primality of Mersenne numbers, 2010.
Jiří Klaška, A Simple Proof of Skula's Theorem on Prime Power Divisors of Mersenne Numbers, J. Int. Seq., Vol. 25 (2022), Article 22.4.3.
Gabriel Lapointe, On finding the smallest happy numbers of any heights, arXiv:1904.12032 [math.NT], 2019.
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012.
Anthony G. Shannon, Hakan Akkuş, Yeşim Aküzüm, Ömür Deveci, and Engin Özkan, A partial recurrence Fibonacci link, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 530-537. See Table 1, p. 531.
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Thesaurus.maths.org, Mersenne Number.
Gérard Villemin's Almanach of Numbers, Nombre de Mersenne.
Eric Wegrzynowski, Nombres de Mersenne. [from Internet Archive Wayback Machine]
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math., Vol. 3 (1892), pp. 265-284.

Cf. A000043, A000668, A046051, A057951-A057958, A100105, A184085, A262153.
Cf. A000040, A000225. - Omar E. Pol, Aug 31 2008
Cf. A002145, A045326.

[2^NthPrime(n)-1: n in [1..30]]; // Vincenzo Librandi, Feb 04 2016

A001348 := n -> 2^(ithprime(n))-1: seq (A001348(n), n=1..18);

Table[2^Prime[n]-1, {n, 20}] (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)

a(n)=1<Charles R Greathouse IV, Jun 10 2011

from sympy import prime
def a(n): return 2**prime(n)-1
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Mar 28 2022

a(n) = 2^A000040(n) - 1, n >= 1. - Wolfdieter Lang, Oct 26 2014
a(n) = A000225(A000040(n)). - Omar E. Pol, Aug 31 2008
A000668(n) = a(A016027(n)). - Omar E. Pol, Jun 29 2012
Sum_{n>=1} 1/a(n) = A262153. - Amiram Eldar, Nov 20 2020
Product_{n>=1} (1 - 1/a(n)) = A184085. - Amiram Eldar, Nov 22 2022

A050217 Super-Poulet numbers: Poulet numbers whose divisors d all satisfy d|2^d-2.

Original entry on oeis.org

341, 1387, 2047, 2701, 3277, 4033, 4369, 4681, 5461, 7957, 8321, 10261, 13747, 14491, 15709, 18721, 19951, 23377, 31417, 31609, 31621, 35333, 42799, 49141, 49981, 60701, 60787, 65077, 65281, 80581, 83333, 85489, 88357, 90751

Offset: 1

Eric W. Weisstein

nonn

Every semiprime in A001567 is in this sequence (see Sierpiński). a(61) = 294409 is the first term having more than two prime factors. See A178997 for super-Poulet numbers having more than two prime factors. - T. D. Noe, Jan 11 2011
Composite numbers n such that 2^d == 2 (mod n) for every d|n. - Thomas Ordowski, Sep 04 2016
Composite numbers n such that 2^p == 2 (mod n) for every prime p|n. - Thomas Ordowski, Sep 06 2016
Composite numbers n = p(1)^e(1)*p(2)^e(2)*...*p(k)^e(k) such that 2^gcd(p(1)-1,p(2)-1,...,p(k)-1) == 1 (mod n). - Thomas Ordowski, Sep 12 2016
Nonsquarefree terms are divisible by the square of a Wieferich prime (see A001220). These include 1194649, 12327121, 5654273717, 26092328809, 129816911251. - Robert Israel, Sep 13 2016
Composite numbers n such that 2^A258409(n) == 1 (mod n). - Thomas Ordowski, Sep 15 2016

W. Sierpiński, Elementary Theory of Numbers, Warszawa, 1964, p. 231.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Super-Poulet Numbers
Wikipedia, Super-Poulet number

A214305 is a subsequence.
A065341 is a subsequence. - Thomas Ordowski, Nov 20 2016
Cf. A001220, A001567, A178997.

filter:= = proc(n)
    not isprime(n) and andmap(p -> 2&^p mod n = 2, numtheory:-factorset(n))
end proc:
select(filter, [seq(i,i=3..10^5,2)]); # Robert Israel, Sep 13 2016

Select[Range[1, 110000, 2], !PrimeQ[#] && Union[PowerMod[2, Rest[Divisors[#]], #]] == {2} & ]

is(n)=if(isprime(n), return(0)); fordiv(n,d, if(Mod(2,d)^d!=2, return(0))); n>1 \\ Charles R Greathouse IV, Aug 27 2016

A135978 Primes p such that 2^p-1 has exactly 2 prime factors.

Original entry on oeis.org

11, 23, 37, 41, 59, 67, 83, 97, 101, 103, 109, 131, 137, 139, 149, 167, 197, 199, 227, 241, 269, 271, 281, 293, 347, 373, 379, 421, 457, 487, 523, 727, 809, 881, 971, 983, 997, 1061, 1063

Offset: 1

Artur Jasinski, Dec 09 2007

a(40)>=1277. - Amiram Eldar, Sep 29 2018

Cf. A000225, A065341, A054723, A134852, A135975, A135976, A135977.

k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d == 2, AppendTo[k, Prime[n]]]], {n, 1, 40}]; k

a(17)-a(37) from Arkadiusz Wesolowski, Jan 26 2012

a(38)-a(39) from Amiram Eldar, Sep 29 2018

A135979 Indices n such that 2^prime(n)-1 has exactly 2 distinct prime factors.

Original entry on oeis.org

5, 9, 12, 13, 17, 19, 23, 25, 26, 27, 29, 32, 33, 34, 35, 39, 45, 46, 49, 53, 57, 58, 60, 62, 69, 74, 75, 82, 88, 93, 99, 129, 140, 152, 164, 166, 168, 178, 179

Offset: 1

Artur Jasinski, Dec 09 2007

a(40)>=206. - Amiram Eldar, Sep 29 2018

Cf. A000225, A065341, A054723, A134852, A135975, A135976, A135977, A135978.

k = {}; Do[If[ ! PrimeQ[2^Prime[n] - 1], c = FactorInteger[2^Prime[n] - 1]; d = Length[c]; If[d == 2, AppendTo[k, n]]], {n, 1, 40}]; k
Select[Range[40],PrimeNu[2^Prime[#]-1]==2&] (* Harvey P. Dale, Jul 07 2013 *)

Equals {k: A001221(A001348(k)) = 2}. a(n) = A049084(A135978(n)). - R. J. Mathar, May 03 2008

Edited by R. J. Mathar, May 03 2008
a(17)-a(34) from Donovan Johnson, Jun 14 2009
a(35)-a(39) from Amiram Eldar, Sep 29 2018

cp's OEIS Frontend

A135975 Number of prime factors (without multiplicity) in Mersenne composites A065341.

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A135977 Mersenne composites (A065341) with exactly 3 prime factors.

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A135976 Mersenne composites (A065341) with exactly 2 prime factors.

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A145097 a(n) is the largest proper divisor of the Mersenne composite A065341(n).

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A136032 Number of prime factors (with multiplicity) of Mersenne composites (A065341).

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A135386 Mersenne composites A065341 with 4 or more prime factors.

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A001348 Mersenne numbers: 2^p - 1, where p is prime.

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A050217 Super-Poulet numbers: Poulet numbers whose divisors d all satisfy d|2^d-2.