A046800 -id:A046800 - cp's OEIS frontend

A172290 Prime divisors of 2^1092-1, listed with multiplicities.

3, 3, 5, 7, 7, 13, 13, 29, 43, 53, 79, 113, 127, 157, 313, 337, 547, 911, 1093, 1093, 1249, 1429, 1613, 2731, 3121, 4733, 5419, 8191, 14449, 21841, 121369, 224771, 503413, 1210483, 1948129, 22366891, 108749551, 112901153, 23140471537, 25829691707, 105310750819, 467811806281, 4093204977277417, 8861085190774909, 556338525912325157, 86977595801949844993, 275700717951546566946854497, 292653113147157205779127526827, 3194753987813988499397428643895659569

Offset: 1

Artur Jasinski, Jan 30 2010

Up to now only two primes p such that p^2 divide 2^(p-1)-1 are known (these two are Wieferich primes, see A001220).
The sequence is finite with A001222(2^1092-1) = 49 terms; A001221(2^1092-1) = 45. - Reinhard Zumkeller, May 14 2010
Terms appearing more than once (in fact twice) are 3, 7, 13, and 1093.

Reinhard Zumkeller, Table of n, a(n) for n = 1..49 (complete sequence)
Dario A. Alpern, Factorization using the Elliptic Curve Method.

Cf. A001220, A172291, A177855, A046051, A046800, A242715.

Missing terms a(34) and a(35) inserted by Reinhard Zumkeller, May 14 2010

Definition clarified and terms corrected by Joerg Arndt, Apr 25 2011

A067886 Numbers k such that 2^k+1 and 2^k-1 have the same number of distinct prime factors.

Original entry on oeis.org

2, 3, 6, 9, 11, 14, 15, 18, 21, 23, 27, 29, 33, 42, 47, 51, 53, 54, 57, 69, 71, 73, 74, 81, 82, 86, 95, 101, 105, 111, 113, 114, 115, 121, 129, 130, 138, 141, 142, 165, 167, 169, 179, 181, 199, 203, 209, 213, 230, 233, 235, 243, 250, 255, 258, 277, 279, 306, 307

Offset: 1

Benoit Cloitre, Mar 02 2002

nonn

Numbers k such that omega(2^k+1) = omega(2^k-1).

Amiram Eldar, Table of n, a(n) for n = 1..141

Cf. A001221, A046799, A046800.

[k: k in [2..307] | #PrimeDivisors(2^k-1) eq #PrimeDivisors(2^k+1) ]; // Marius A. Burtea, Feb 13 2020

sndpQ[n_]:=Module[{c=2^n},PrimeNu[c+1]==PrimeNu[c-1]]; Select[Range[ 250], sndpQ] (* Harvey P. Dale, Feb 04 2016 *)

isok(k) = omega(2^k-1) == omega(2^k+1); \\ Michel Marcus, Feb 13 2020

More terms from Rick L. Shepherd, May 14 2002

More terms from Amiram Eldar, Feb 13 2020

A335432 Number of anti-run permutations of the prime indices of Mersenne numbers A000225(n) = 2^n - 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 6, 2, 6, 2, 36, 1, 6, 6, 24, 1, 24, 1, 240, 6, 24, 2, 1800, 6, 6, 6, 720, 6, 1800, 1, 120, 24, 6, 24, 282240, 2, 6, 24, 15120, 2, 5760, 6, 5040, 720, 24, 6, 1451520, 2, 5040, 120, 5040, 6, 1800, 720, 40320, 24, 720, 2, 1117670400, 1, 6, 1800, 5040, 6

Offset: 1

Gus Wiseman, Jul 02 2020

nonn

An anti-run is a sequence with no adjacent equal parts.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(1) = 1 through a(10) = 6 permutations:
  ()  (2)  (4)  (2,3)  (11)  (2,4,2)  (31)  (2,3,7)  (21,4)  (11,2,5)
                (3,2)                       (2,7,3)  (4,21)  (11,5,2)
                                            (3,2,7)          (2,11,5)
                                            (3,7,2)          (2,5,11)
                                            (7,2,3)          (5,11,2)
                                            (7,3,2)          (5,2,11)

Andrew Howroyd, Table of n, a(n) for n = 1..200

The version for factorial numbers is A335407.
Anti-run compositions are A003242.
Anti-run patterns are A005649.
Permutations of prime indices are A008480.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Strict permutations of prime indices are A335489.
Cf. A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335125.
Mersenne numbers: A000225, A046051, A046800, A046801, A049093, A059305, A159611, A325610, A325611, A325612, A325625.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[2^n-1]],!MatchQ[#,{_,x_,x_,_}]&]],{n,0,30}]

\\ See A335452 for count.
a(n) = {count(factor(2^n-1)[,2])} \\ Andrew Howroyd, Feb 03 2021

a(n) = A335452(A000225(n)).

Terms a(51) and beyond from Andrew Howroyd, Feb 03 2021

A283364 Numbers m such that both numbers 2^m +- 1 have at most 2 distinct prime factors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 17, 19, 23, 31, 61, 101, 127, 167, 199, 347

Offset: 1

Vladimir Shevelev, Mar 06 2017

If a(n) > 9 then a(n) is prime. Proof: If k = 2*m > 9 then 2^(2*m)-1 has at least 3 factors; being 3, (2^m - 1) / 3 and 2^m + 1 which excludes even numbers > 9.
If k = 2*m + 1 > 9 is not prime then k = p*q, q, p > 3 so 2^(p*q) + 1 is divisible by 3, 2^p + 1 and 2^q + 1. If p = q then 2^(p^2) + 1 is divisible by 3, 2^p + 1 and (2^p^2 + 1) / (2^p + 1) > 2^p + 1. Which excludes odd composite numbers > 9 and completes the proof. [comments reworded by David A. Corneth, Nov 23 2019]
Any further terms are > 1122. - Lucas A. Brown, Oct 21 2024

Cf. A001221, A046799, A046800, A067886.

Select[Range@ 200, Times @@ Boole@ Map[PrimeNu@ # <= 2 &, 2^# + {-1, 1}] == 1 &] (* Michael De Vlieger, Mar 06 2017 *)
Select[Range[350],Max[PrimeNu[2^#+{1,-1}]]<3&] (* Harvey P. Dale, Dec 23 2017 *)

isok(n) = omega(2^n+1)<=2 && omega(2^n-1)<=2;
for(n=1, 347, if(isok(n)==1, print1(n,", "))); \\ Indranil Ghosh, Mar 06 2017

More terms from Peter J. C. Moses, Mar 06 2017

A337811 Numbers k such that the number of distinct prime factors of 2^k - 1 is less than the corresponding count for 2^k + 1.

Original entry on oeis.org

1, 5, 7, 13, 17, 19, 25, 26, 31, 34, 35, 37, 38, 41, 46, 49, 59, 61, 62, 65, 67, 77, 78, 83, 85, 89, 91, 93, 97, 98, 103, 107, 109, 118, 122, 123, 125, 127, 131, 133, 134, 137, 139, 143, 145, 147, 149, 153, 157, 170, 173, 175, 177, 185, 186, 189, 193, 194, 195

Offset: 1

Hugo Pfoertner, Sep 23 2020

nonn

Cf. A046799, A046800, A067886, A337810, A337812, A337813.

Select[Range[200],PrimeNu[2^#-1]Harvey P. Dale, Nov 04 2023 *)

PARI
```
for(n=1,200,if(omega(2^n-1)
				
```

A337813 Numbers k such that the number of distinct prime factors of 2^k - 1 is greater than the corresponding count for 2^k + 1.

Original entry on oeis.org

4, 8, 10, 12, 16, 20, 22, 24, 28, 30, 32, 36, 39, 40, 43, 44, 45, 48, 50, 52, 55, 56, 58, 60, 63, 64, 66, 68, 70, 72, 75, 76, 79, 80, 84, 87, 88, 90, 92, 94, 96, 99, 100, 102, 104, 106, 108, 110, 112, 116, 117, 119, 120, 124, 126, 128, 132, 135, 136, 140, 144

Offset: 1

Hugo Pfoertner, Sep 23 2020

nonn

Cf. A046799, A046800, A067886, A337810, A337811, A337812.

for(n=1,150,if(omega(2^n-1)>omega(2^n+1),print1(n,", ")))

A152057 Sum of the distinct prime factors of 2^n-1.

Original entry on oeis.org

0, 0, 3, 7, 8, 31, 10, 127, 25, 80, 45, 112, 28, 8191, 173, 189, 282, 131071, 102, 524287, 91, 471, 798, 178528, 286, 2433, 10925, 262737, 320, 3425, 534, 2147483647, 65819, 599598, 174765, 123150, 266, 616318400, 699053, 129646, 61789, 164524720, 5936

Offset: 0

Roger L. Bagula, Nov 22 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 0..1206 (terms 0..500 from Robert Israel)

Cf. A000225, A008472, A046800, A075708.

Row sums of A060443.

sopf:= n -> convert(numtheory:-factorset(n),`+`):
seq(sopf(2^n-1),n=0..100); # Robert Israel, Jan 14 2021

Table[Sum[FactorInteger[2^n - 1][[m]][[1]], {m, 1, Length[FactorInteger[2^n - 1]]}], {n, 0, 50}]

a(n) = A008472(A000225(n)). - Robert Israel, Jan 14 2021

Edited by N. J. A. Sloane, Nov 26 2008

A257861 Numbers n such that d(m) - f(m) >= n/2^f(m), where m = 2^n - 1, d(m) is the number of distinct prime factors of m, and f(m) is the number of Fermat primes less than or equal to 65537 that divide m.

Original entry on oeis.org

24, 48, 64, 72, 80, 96, 112, 128, 144, 160, 192, 224, 240, 288, 320, 336, 352, 384, 416, 448, 480, 576, 672, 800, 864, 960, 1056, 1440

Offset: 1

Arkadiusz Wesolowski, Jul 16 2015

For every n there exists a Sierpiński/Riesel number with modulus a(n).

Wikipedia, Covering set

Cf. A019434, A046800.

lista(nn) = {vfp = [3, 5, 17, 257, 65537]; for(n = 1, nn, m = 2^n-1; dm = omega(m); fm = sum(k=1, #vfp, (m % vfp[k]) == 0); if (dm - fm >= n/2^fm, print1(n, ", ")););} \\ Michel Marcus, Jul 20 2015

A152058 Primes in A152057 in the order of their appearance.

Original entry on oeis.org

3, 7, 31, 127, 8191, 173, 131071, 524287, 2147483647, 699053, 3313, 13271393, 2305843009213693951, 209659, 9361975431089, 10670593, 618970019642690137449562111, 18861569, 23253381919, 162259276829213363391578010288127, 177722254122587407, 170141183460469231731687303715884105727

Offset: 1

Roger L. Bagula, Nov 22 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..34

Cf. A046800, A152057.

Table[If[PrimeQ[s = Total[FactorInteger[2^n - 1][[;; , 1]]]], s, Nothing], {n, 1, 50}] (* Amiram Eldar, Jun 07 2025 *)

Edited by N. J. A. Sloane, Nov 26 2008

Definition revised, offset corrected and more terms added by Amiram Eldar, Jun 07 2025

A244453 Prime factors of 2^A054723(n)-1, ordered by increasing n, then by increasing size of the factors.

Original entry on oeis.org

23, 89, 47, 178481, 233, 1103, 2089, 223, 616318177, 13367, 164511353, 431, 9719, 2099863, 2351, 4513, 13264529, 6361, 69431, 20394401, 179951, 3203431780337, 193707721, 761838257287, 228479, 48544121, 212885833

Offset: 1

Felix Fröhlich, Jun 28 2014

Subsequence of A060443.

Prime factors of composite Mersenne numbers; A089162 with the Mersenne primes A000668 removed. - Jens Kruse Andersen, Jul 11 2014

			A054723(1) = 11. 2^11-1 = 2047 = 23*89. - _Jens Kruse Andersen_, Jul 11 2014
Triangle begins:
23, 89;
47, 178481;
233, 1103, 2089;
223, 616318177;
13367, 164511353;
431, 9719, 2099863;
2351, 4513, 13264529;
6361, 69431, 20394401;

Jens Kruse Andersen, Table of n, a(n) for n = 1..577
Sergey Nikitin, Euler-Fermat algorithm and some of its applications, 2018.
Sam Wagstaff, The Cunningham Project

Cf. A003260, A016047, A046800, A089162.

Map[FactorInteger, Select[2^Prime@Range@20 - 1, CompositeQ]][[All, All, 1]] // Flatten (* Michael De Vlieger, Nov 20 2018 *)

forprime(n=1, 100, m=2^n-1; if(!isprime(m), f=factor(m); for(i=1, #f~, print1(f[i,1]", ")))) \\ Jens Kruse Andersen, Jul 11 2014

cp's OEIS Frontend

A172290 Prime divisors of 2^1092-1, listed with multiplicities.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A067886 Numbers k such that 2^k+1 and 2^k-1 have the same number of distinct prime factors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A335432 Number of anti-run permutations of the prime indices of Mersenne numbers A000225(n) = 2^n - 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A283364 Numbers m such that both numbers 2^m +- 1 have at most 2 distinct prime factors.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A337811 Numbers k such that the number of distinct prime factors of 2^k - 1 is less than the corresponding count for 2^k + 1.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

A337813 Numbers k such that the number of distinct prime factors of 2^k - 1 is greater than the corresponding count for 2^k + 1.

Views

Author

Keywords

Crossrefs

Programs

PARI

A152057 Sum of the distinct prime factors of 2^n-1.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A257861 Numbers n such that d(m) - f(m) >= n/2^f(m), where m = 2^n - 1, d(m) is the number of distinct prime factors of m, and f(m) is the number of Fermat primes less than or equal to 65537 that divide m.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A152058 Primes in A152057 in the order of their appearance.