A046051 -id:A046051 - cp's OEIS frontend

A337812 Numbers k such that the number of prime factors, counted with multiplicity, of 2^k - 1 is greater than the corresponding count for 2^k + 1.

4, 6, 8, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 43, 44, 48, 52, 54, 56, 58, 60, 64, 66, 68, 70, 72, 76, 79, 80, 84, 87, 88, 90, 92, 94, 96, 100, 102, 104, 106, 108, 110, 112, 114, 116, 117, 119, 120, 124, 126, 128, 132, 136, 138, 140, 144, 146, 148, 151

Offset: 1

Hugo Pfoertner, Sep 23 2020

nonn

Cf. A046051, A054992, A309942, A337810, A337811, A337813.

Select[Range[152],PrimeOmega[2^#-1]>PrimeOmega[2^#+1]&] (* Harvey P. Dale, Dec 31 2022 *)

for(n=1,160,if(bigomega(2^n-1)>bigomega(2^n+1),print1(n,", ")))

A109472 Cumulative sum of primes p such that 2^p - 1 is a Mersenne prime.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Aug 28 2005

nonn

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.

			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).

Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Gord Palameta)

Cf. A000043, A000668 for the Mersenne primes, A001348, A046051, A057951-A057958.

Accumulate[Select[Range[3000], PrimeQ[2^# - 1] &]] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)
Accumulate@ MersennePrimeExponent@ Range@ 45 (* Michael De Vlieger, Jul 22 2018 *)

a(n) = Sum_{i=1..n} A000043(i).

a(38)-a(47) from Gord Palameta, Jul 21 2018

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

A193295 Number of prime divisors (with multiplicity) of n^2 - 1.

Original entry on oeis.org

1, 3, 2, 4, 2, 5, 3, 5, 3, 5, 2, 5, 3, 6, 3, 7, 2, 6, 3, 5, 3, 6, 3, 6, 5, 5, 4, 6, 2, 8, 3, 7, 4, 6, 3, 6, 3, 6, 3, 7, 2, 6, 4, 5, 4, 7, 3, 8, 4, 6, 3, 7, 3, 8, 4, 6, 3, 6, 2, 6, 4, 8, 5, 9, 3, 6, 3, 6, 3, 8, 2, 7, 4, 5, 5, 6, 3, 8, 5, 7, 5, 6, 3, 6, 4, 6

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 22 2011

nonn

T. D. Noe, Table of n, a(n) for n = 2..10000

Cf. A046051, A054992, A054990, A054991, A082863.

Table[PrimeOmega[n^2 - 1], {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

a(n)=bigomega(n^2-1) \\ Charles R Greathouse IV, Jul 30 2011

A309942 Numbers k such that 2^k - 1 and 2^k + 1 have the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

2, 10, 11, 14, 21, 23, 29, 39, 47, 50, 53, 55, 63, 71, 73, 74, 75, 82, 86, 95, 101, 105, 113, 115, 121, 142, 147, 150, 167, 169, 179, 181, 182, 190, 199, 203, 209, 233, 235, 253, 277, 285, 303, 307, 311, 317, 335, 337, 339, 342, 343, 347, 349, 353, 355, 358

Offset: 1

Hugo Pfoertner, Aug 24 2019

nonn

			a(1) = 2: 2^2 - 1 = 3 and 2^2 + 1 are both prime,
a(2) = 10: 2^10 - 1 = 1023 = 3 * 11 * 31 and 2^10 + 1 = 1025 = 5^2 * 41 both have 3 prime factors.

Cf. A000051, A000225, A046051, A054992, A067886.

[m:m in [2..400]| &+[p[2]: p in Factorization(2^m-1)] eq &+[p[2]: p in Factorization(2^m+1)]]; // Marius A. Burtea, Aug 24 2019

Select[Range[200], PrimeOmega[2^# - 1 ] == PrimeOmega[2^# + 1 ] &] (* Amiram Eldar, Aug 24 2019 *)

for(k=1, 209, my(f=bigomega(2^k-1),g=bigomega(2^k+1));if(f==g,print1(k,", ")))

More terms from Amiram Eldar, Aug 24 2019

A135981 Number of distinct prime factors of A135972(n).

Original entry on oeis.org

0, 2, 2, 3, 2, 3, 2, 4, 3, 3, 4, 4, 5, 3, 4, 2, 6, 3, 3, 3, 6, 3, 6, 5, 4, 3, 4, 8, 2, 3, 4, 7, 2, 6, 3, 7, 6, 4, 3, 9, 2, 7, 5, 7, 3, 6, 6, 8, 4, 6, 2, 11, 3, 6, 7, 3, 8, 2, 7, 4, 9, 3, 12, 3, 5, 7, 7, 4, 7, 3, 9, 6, 5, 2, 12, 3, 5, 6, 10, 11, 5, 9, 3, 6, 5, 12, 2, 5, 8, 12

Offset: 2

Artur Jasinski, Dec 09 2007

nonn

			A135972(3) = 15 = 3*5 which has a(3)=2 distinct prime factors.

Amiram Eldar, Table of n, a(n) for n = 2..1193

Cf. A046051, A046053, A022307, A135972.

k = {}; Do[If[ ! PrimeQ[2^n - 1], c = FactorInteger[2^n - 1]; d = Length[c]; AppendTo[k, d]], {n, 1, 100}]; k

a(n) = A001221(A135972(n)) .

Offset set to 2, definition shortened - R. J. Mathar, Oct 01 2009

A140745 Smallest prime p such that the Mersenne number A000225(p) = 2^p - 1 has exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

2, 11, 29, 157, 113, 223, 491, 431, 397

Offset: 1

Lekraj Beedassy, Jul 12 2008

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 223, pp 63-4, Ellipse Paris 2008.

Cf. A000225, A001222, A046051, A136033.

a[n_]:=Module[{p=0},Until[PrimeOmega[2^Prime[p]-1]==n,p++];Prime[p]];Array[a,6] (* James C. McMahon, Jul 14 2025 *)

a(n) = forprime(p=2, oo, if(bigomega(2^p-1)==n, return(p))); \\ Jinyuan Wang, Aug 10 2021

A155900 Numbers k such that Omega(k) = Omega(2^k-1), where Omega(k) is the number of prime factors of k counted with multiplicity (A001222).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 13, 16, 17, 19, 27, 31, 32, 49, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217

Offset: 1

M. F. Hasler, Feb 01 2009

Mersenne prime exponents A000043 are a subsequence, with Omega(p)=Omega(2^p-1)=1.

Davar55 et al., "Mersenne-Like", discussion on mersenneforum.org, Jan 27 2009.

Cf. A000043, A000225, A001222, A046051.

Select[Range[200],PrimeOmega[#]==PrimeOmega[2^#-1]&] (* Harvey P. Dale, Apr 21 2012 *)

for( i=1,999, bigomega(2^i-1)==bigomega(i) & print1(i","))

{n: A046051(n) = A001222(n)}. - R. J. Mathar, Mar 14 2009

a(21)-a(22) from Amiram Eldar, Feb 23 2021

a(23)-a(26) from Daniel Suteu, Jan 21 2023

A215896 Largest k = 2^(m - 1)*(2^m - 1) such that bigomega(k) = n or 0 if no such k exists.

Original entry on oeis.org

1, 0, 6, 28, 0, 496, 0, 8128, 2016, 0, 130816, 0, 2096128, 33550336, 0, 0, 134209536, 8589869056, 0, 137438691328, 0, 0, 0, 34359607296, 35184367894528, 8796090925056, 0, 562949936644096, 2251799780130816, 9007199187632128, 140737479966720, 2305843008139952128, 0

Offset: 1

Gerasimov Sergey, Aug 25 2012

nonn

Largest k = 2^(m-1)*(2^m-1) such that bigomega(k) = prime(n) or 0 if no such k exists (other version): 6, 28, 496, 8128, 0, 0, 8589869056, 137438691328, 34359607296, 9007199187632128, 2305843008139952128, 0, ...
Mersenne exponents (A000043): numbers n such that omega(2^(n-1)*(2^n-1)) = 2, or bigomega(2^(n-1)*(2^n-1)) = n, or tau(2^(n-1)*(2^n-1)) = 2n, or sigma(2^(n-1)*(2^n-1)) = 2^n*(2^n-1).
Smallest k = 2^(m-1)*(2^m-1) such that bigomega(k) = n or 0 if no such k exists : 1, 0, 6, 28, 0, 120, 0, 8128, 2016, 0, 32640, 0, 523776, 33550336, 0, 0, 8386560, 536854528, 0, 2147450880, 0, 0, 0, 34359607296, 2199022206976, 549755289600, 0, 562949936644096, 2251799780130816,...

			a(0) = 1 because 2^(1-1)*(2^1-1) = 1 and A001222(1) = 0,
a(2) = 6 because 2^(2-1)*(2^2-1) = 6 and A001222(6) = 2,
a(3) = 28 because 2^(3-1)*(2^3-1) = 28 and A001222(28) = 3,
a(5) = 496 because 2^(4-1)*(2^4-1) = 120, 2^(5-1)*(2^5-1) = 496 and A001222(120) = A001222(496) = 5, 496 > 120.
a(7) = 8128 because 2^(7-1)*(2^7-1) = 8128 and A001222(8128) = 7,
a(8) = 2016 because 2^(6-1)*(2^6-1) = 2016 and A001222(2016) = 8,
a(10) = 130816 because 2^(8-1)*(2^8-1) = 32640, 2^(9-1)*(2^9-1) = 130816 and A001222(32640) = A001222(130816) = 10, 130816 > 32640.

Cf. A000396, A006516, A046051, A144858.

A215896 := proc(n)
      local m,k;
      for m from n+2 by -1 do
        k := 2^(m-1)*(2^m-1) ;
        if k < 0 then
            return 0 ;
        end if;
        if numtheory[bigomega](k) = n then
            return k ;
        end if;
    end do:
end proc: # R. J. Mathar, Sep 11 2012

A226116 Numbers k such that one of 2^k-1 or 2^k+1 is semiprime, but not both.

Original entry on oeis.org

3, 4, 5, 6, 7, 9, 12, 13, 17, 19, 20, 28, 31, 32, 37, 40, 41, 43, 49, 59, 61, 64, 67, 79, 83, 92, 97, 103, 104, 109, 127, 128, 131, 137, 139, 148, 149, 191, 197, 227, 241, 256, 269, 271, 281, 293, 313, 356, 373, 379, 421, 457, 487, 523, 596, 692, 701, 727, 809, 881, 971, 983, 997, 1004, 1061, 1063

Offset: 1

Irina Gerasimova, May 28 2013

			2^3-1=7 is not a semiprime but 2^3+1 =9 is, so 3 is in the sequence.
2^4-1 =15 is a semiprime but 2^4+1 =17 is not, so 4 is in the sequence.
2^8-1 =255 is a 3-prime (not a 2-prime) and 2^8+1 =257 is a prime (not a 2-prime), so 8 is not in the sequence.

Cf. A001358, A046051, A054992, A092558, A092559.

isok(n) = {nbm = bigomega(2^n-1); nbp = bigomega(2^n+1); return (((nbm == 2) || (nbp == 2)) && ! ((nbm == 2) && (nbp == 2)));} \\ Michel Marcus, Aug 23 2013

Original sequence of 4 small numbers replaced by a wider sequence. - R. J. Mathar, Jun 13 2013

cp's OEIS Frontend

A337812 Numbers k such that the number of prime factors, counted with multiplicity, of 2^k - 1 is greater than the corresponding count for 2^k + 1.

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A109472 Cumulative sum of primes p such that 2^p - 1 is a Mersenne prime.

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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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Maple

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A193295 Number of prime divisors (with multiplicity) of n^2 - 1.

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A309942 Numbers k such that 2^k - 1 and 2^k + 1 have the same number of prime factors, counted with multiplicity.

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Magma

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A135981 Number of distinct prime factors of A135972(n).

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A140745 Smallest prime p such that the Mersenne number A000225(p) = 2^p - 1 has exactly n prime factors (counted with multiplicity).

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A155900 Numbers k such that Omega(k) = Omega(2^k-1), where Omega(k) is the number of prime factors of k counted with multiplicity (A001222).

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