A136030 -id:A136030 - cp's OEIS frontend

A065341 Mersenne composites: 2^prime(m) - 1 is not a prime.

2047, 8388607, 536870911, 137438953471, 2199023255551, 8796093022207, 140737488355327, 9007199254740991, 576460752303423487, 147573952589676412927, 2361183241434822606847, 9444732965739290427391

Offset: 1

Labos Elemer, Oct 30 2001

nonn

For the number of prime factors in a(n) see A135975. For indices of primes n in composite 2^prime(n)-1 see A135980. For smallest prime divisors of Mersenne composites see A136030. For largest prime divisors of Mersenne composites see A136031. For largest divisors see A145097. - Artur Jasinski, Oct 01 2008

All the terms are Fermat pseudoprimes to base 2 (A001567). For a proof see, e.g., Jaroma and Reddy (2007). - Amiram Eldar, Jul 24 2021

			2^11 - 1 = 2047 = 23*89.

Muniru A Asiru, Table of n, a(n) for n = 1..110
John H. Jaroma and Kamaliya N. Reddy, Classical and alternative approaches to the Mersenne and Fermat numbers, The American Mathematical Monthly, Vol. 114, No. 8 (2007), pp. 677-687.

Cf. A054723, A000043, A000668, A001348, A001567.

Cf. A135975, A135980, A145097, A136031. - Artur Jasinski, Oct 01 2008

A065341 := proc(n) local i;
i := 2^(ithprime(n))-1:
if (not isprime(i)) then
   RETURN (i)
fi: end: seq(A065341(n), n=1..21); # Jani Melik, Feb 09 2011

Select[Table[2^Prime[n]-1,{n,30}],!PrimeQ[#]&] (* Harvey P. Dale, May 06 2018 *)

a(n) = 2^A054723(n) - 1.

A136031 Largest prime factor of composite Mersenne numbers.

Original entry on oeis.org

89, 178481, 2089, 616318177, 164511353, 2099863, 13264529, 20394401, 3203431780337, 761838257287, 212885833, 9361973132609, 1113491139767, 57912614113275649087721, 13842607235828485645766393, 341117531003194129, 3976656429941438590393

Offset: 1

Artur Jasinski, Dec 11 2007

nonn

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

Cf. A049479, A016047, A003260, A136030.

FactorInteger[2^#-1][[-1,1]]&/@Select[Prime[Range[40]],!PrimeQ[2^#-1]&] (* Harvey P. Dale, May 05 2012 *)
With[{c=Complement[Prime[Range[PrimePi[200]]],MersennePrimeExponent[ Range[ 15]]]},Table[ FactorInteger[ 2^p-1][[-1,1]],{p,c}]] (* Harvey P. Dale, Sep 06 2021 *)

a(n) >= A089158(n). - R. J. Mathar, May 01 2008

Corrected by Harvey P. Dale, May 05 2012

A136033 a(n) = smallest number k such that number of prime factors of 2^k-1 is exactly n (counted with multiplicity).

Original entry on oeis.org

2, 4, 6, 16, 12, 18, 24, 40, 54, 36, 102, 110, 60, 72, 108, 140, 120, 156, 144, 200, 216, 210, 240, 180, 456, 288, 336, 300, 396, 480, 882, 360, 468, 700

Offset: 1

Artur Jasinski, Dec 11 2007

Cf. A000225, A003260, A016047, A046051, A049479, A088863, A136030, A136031.

N:= 24: # to get a(1) to a(N)
unknown:= N:
for k from 2 while unknown > 0 do
  q:= numtheory:-bigomega(2^k-1);
  if q <= N and not assigned(A[q]) then
     A[q]:= k;
     unknown:= unknown - 1;
  fi
od:
seq(A[i],i=1..N); # Robert Israel, Oct 24 2014

Module[{nn=250,tbl},tbl=Table[{k,PrimeOmega[2^k-1]},{k,nn}];Table[SelectFirst[tbl,#[[2]]==n&],{n,24}]][[;;,1]] (* The program generates the first 24 terms of the sequence. *)  (* Harvey P. Dale, May 25 2025 *)

a(n) = {k = 1; while(bigomega(2^k-1) != n, k++); k;} \\ Michel Marcus, Nov 04 2013

a(15)-a(20) from Michel Marcus, Nov 04 2013
a(21)-a(24) from Derek Orr, Oct 23 2014
a(25)-a(34) from Jinyuan Wang, Jun 07 2019

A136034 a(n) = smallest number k such that number of distinct prime factors of 2^k-1 is exactly n.

Original entry on oeis.org

1, 2, 4, 8, 12, 20, 24, 40, 36, 48, 88, 60, 72, 150, 132, 120, 156, 144, 200, 204, 210, 180, 324, 476, 288, 300, 432, 396, 480, 360, 468, 576, 700, 504, 420, 648, 540, 660, 792, 720

Offset: 0

Artur Jasinski, Dec 11 2007

First occurrence of n in A046800.

Cf. A000225, A003260, A016047, A046051, A046800, A049479, A088863, A136030, A136031, A136032, A136033.

With[{pn1=PrimeNu[2^Range[800]-1]},Table[Position[pn1,n,1,1],{n,0,40}]]//Flatten (* Harvey P. Dale, Jan 10 2025 *)

a(n) = my(k=1); while (omega(2^k-1) != n, k++); k; \\ Michel Marcus, Jan 09 2023

More terms from Julián Aguirre, Feb 04 2013
a(31)-a(39) from Chai Wah Wu, Oct 03 2019
a(0) = 1 inserted by Michel Marcus, Jan 09 2023

A241973 Prime exponents of composite Mersenne numbers in the order of the magnitude of the smallest prime factor.

Original entry on oeis.org

11, 23, 83, 37, 29, 131, 179, 191, 43, 73, 239, 251, 359, 419, 431, 443, 491, 659, 683, 233, 719, 743, 911, 1019, 1031, 1103, 47, 397, 1223, 79, 461, 1439, 1451, 1499, 1511, 1559, 1583, 557, 113, 577, 601, 1811, 1931, 2003, 2039, 2063, 761, 2339, 2351, 2399

Offset: 1

J. Lowell, May 03 2014

nonn

Terms are the same as A054723, but in a different order.
If p is a prime and 2^p-1 is composite, each prime factor of 2^p-1 will be of the form kp+1 for some integer k. Thus, the smallest prime factor of 2^p-1 cannot be smaller than p.
The corresponding smallest prime factors are: 23, 47, 167, 223, 233, 263, 359, 383, 431, 439, 479, 503, 719, 839, 863, 887, 983, ....

			83 comes before 37 because 167 (the smallest prime factor of 2^83-1) < 223 (the smallest prime factor of 2^37-1).

Cf. A054723, A136030.

lista() = {vi = readvec("b054723.txt"); vm = vector(#vi, i, 2^vi[i]-1); p = 2; nbf = 0; while ( nbf != #vm, i = 1; while (!(i>#vm) && (!vm[i] || (vm[i] % p)), i++); if (i <= #vm, print1(vi[i], ", "); vm[i] = 0; nbf ++;); p = nextprime(p+1););} \\ Michel Marcus, May 14 2014

More terms from Michel Marcus, May 14 2014

cp's OEIS Frontend

A065341 Mersenne composites: 2^prime(m) - 1 is not a prime.

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A136031 Largest prime factor of composite Mersenne numbers.

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A136033 a(n) = smallest number k such that number of prime factors of 2^k-1 is exactly n (counted with multiplicity).

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A136034 a(n) = smallest number k such that number of distinct prime factors of 2^k-1 is exactly n.

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Mathematica

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A241973 Prime exponents of composite Mersenne numbers in the order of the magnitude of the smallest prime factor.

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PARI

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