A122094 -id:A122094 - cp's OEIS frontend

A322568 Integers k such that the least prime factor of 2^k - 1 is not in A122094.

169, 221, 323, 611, 779, 793, 923, 1121, 1159, 1271, 1273, 1349, 1513, 1717, 1829, 1919, 2033, 2077, 2197, 2201, 2413, 2533, 2603, 2759, 2873, 2951, 3097, 3131, 3173, 3193, 3211, 3281, 3379, 3599, 3721, 3757, 3791, 3937, 3953, 4043, 4199, 4223, 4309, 4331

Offset: 1

Jeppe Stig Nielsen, Aug 29 2019

nonn

Clearly, the terms are odd and composite (A071904).

The first term which is itself of form 2^j - 1 is 34359738367 = 2^35 - 1. The least prime factor of 2^34359738367 - 1 is 136463, and the multiplicative order of 2 modulo 136463 is 2201 = 31*71. In A309130, it is asked if a member of A322568 can be of form 2^p - 1 with p prime.

			169 is included because the least prime factor of 2^169-1 is 4057, and the multiplicative order of 2 modulo 4057 is 169 which is not prime. The divisor 4057 is less than the "algebraic" divisor 2^13-1 = 8192 (Mersenne prime).
4199 (= 13*17*19) is included because the least prime factor of 2^4199-1 is 647, and the multiplicative order of 2 modulo 647 is 323 (= 17*19) which is not prime. The divisor 647 is less than the smallest "algebraic" divisor which is 2^13-1 = 8192 (Mersenne prime).
289 is NOT included; its least prime factor is 2^17 - 1.
1073 (= 29*37) is NOT included; its least prime factor is 223, but 223 is a divisor of one of the "algebraic" factors, namely 223 is a divisor of composite Mersenne number 2^37 - 1.

Cf. A000225, A020639, A122094, A309130.

for(k=2,+oo,isprime(k)&&next();forprime(p=3,,if(Mod(2,p)^k-1==0,!isprime(znorder(Mod(2,p)))&&print1(k,", ");next(2))))

A089162 Triangle read by rows formed by the prime factors of Mersenne number 2^prime(n) - 1, n >= 1.

Original entry on oeis.org

3, 7, 31, 127, 23, 89, 8191, 131071, 524287, 47, 178481, 233, 1103, 2089, 2147483647, 223, 616318177, 13367, 164511353, 431, 9719, 2099863, 2351, 4513, 13264529, 6361, 69431, 20394401, 179951, 3203431780337, 2305843009213693951, 193707721, 761838257287

Offset: 1

Cino Hilliard, Dec 06 2003

All factors of Mersenne numbers 2^p - 1, where p is prime, are == 1 (mod p). See the first Caldwell link for a proof of the statement that if q divides M_p = 2^p-1 then q = 2kp + 1 for some integer k. - Comment corrected by Jonathan Sondow, Dec 29 2016

			The 16th Mersenne number 2^53-1 has the three prime factors 6361, 69431, 20394401.
See tail end of second row in the sequence. Each factor is == 1 (mod 53).
Triangle begins:
  3;
  7;
  31;
  127;
  23, 89;
  8191;
  131071;
  524287;
  47, 178481;
  233, 1103, 2089;
  2147483647;
  223, 616318177;
  13367, 164511353;
  431, 9719, 2099863;
  2351, 4513, 13264529;
  6361, 69431, 20394401;

Max Alekseyev, Rows n = 1..197, flattened (rows 1..167 from Jens Kruse Andersen)
R. P. Brent, New factors of Mersenne numbers.
Chris K. Caldwell, Mersenne Primes: History, Theorems and Lists.
Chris K. Caldwell, The Prime Glossary, Mersenne divisor.
Sam Wagstaff, The Cunningham Project.

Cf. A001348, A003260, A016047, A088863.

Cf. A122094 (sorted version of this list).

row[n_]:=First/@FactorInteger[2^Prime[n]-1]; Array[row,19]//Flatten (* Stefano Spezia, May 03 2024 *)

mersenne(b,n,d) = { c=0; forprime(x=2,n, c++; y = b^x-1; f=factor(y); v=component(f,1); ln = length(v); if(ln>=d,print1(v[d]",")); ) }

Definition corrected by Max Alekseyev, Jul 25 2023

A263686 Smallest prime factor of double Mersenne numbers.

Original entry on oeis.org

7, 127, 2147483647, 170141183460469231731687303715884105727, 338193759479, 231733529, 62914441, 295257526626031

Offset: 1

Arkadiusz Wesolowski, Oct 23 2015

A double Mersenne number is a Mersenne number of the form 2^(2^p - 1) - 1, where p is a Mersenne exponent (A000043).
From M. F. Hasler, Feb 28 2025: (Start)
The prime factors of Mersenne numbers 2^q - 1 must be of the form 2*q*k + 1.
The four smallest double Mersenne numbers (p = 2, 3, 5, 7 => q = 3, 7, 31, 127) are prime, so their smallest prime factor is equal to themselves, a(n) = M(q). This is equivalent to k = (2^(q-1)-1)/q, which is almost as large as M(q) itself: k = 1, 9 and 34636833 for the first three terms, and for q = 127, k has just three digits less than M(q) = a(4) itself. The prime p = 11 is not a Mersenne exponent.
The fifth term, a(5) = 2*(2^13-1)*k + 1 with k = 20644229 (which is prime) is the first proper divisor of the respective M(q), as are the next three, corresponding to p = 17, 19 and 31.
For p = 61, M(q) has 694127911065419642 digits, and so far no factor is known, but it is known that it has no factor less than 10^36. (End)

Double Mersennes Prime Search, History
Wikipedia, Double Mersenne number

Cf. A000043, A000668, A001348, A020639, A049479, A077586, A122094. Subsequence of A016047. Subsequence of A309130.

forprime(p=2,,q=2^p-1; !ispseudoprime(q) && next(); if(ispseudoprime(2^q-1), print1(2^q-1,", ");next()); forstep(r=2*q+1,+oo,2*q, !ispseudoprime(r) && next(); if(Mod(2,r)^q-1 == 0, print1(r,", ");next(2)))) \\ Jeppe Stig Nielsen, Aug 28 2019

a(n) = spf(MM(A000043(n))) = A049479(A000668(n)), where spf = A020639 is the smallest prime factor, A049479 = spf o M, M(p) = 2^p-1 = A000225(p), MM = M o M = A077585, A000668(n) = M(A000043(n)), A000043 are the Mersenne prime exponents. - M. F. Hasler, Mar 01 2025

A277048 Primes p such that the multiplicative order of 3 modulo p is prime.

Original entry on oeis.org

11, 13, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 431, 467, 479, 503, 563, 587, 683, 719, 839, 863, 887, 983, 1019, 1091, 1093, 1187, 1223, 1283, 1307, 1319, 1367, 1439, 1487, 1511, 1523, 1583, 1597, 1619, 1669, 1823, 1871, 1907, 2027

Offset: 1

Vincenzo Librandi, Oct 28 2016

nonn

Odd primes that divide 3^p-1 for some prime p. - Robert Israel, Nov 14 2016

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A122094.

[p: p in PrimesInInterval(2, 4000) | IsPrime(Modorder(3, p))];

select(p -> isprime(p) and isprime(numtheory:-order(3, p)), [seq(p, p=5..10000, 2)]); # Robert Israel, Nov 14 2016

Select[Prime@ Range@ 310, PrimeQ@ MultiplicativeOrder[3, #] &] (* Michael De Vlieger, Oct 28 2016 *)

isok(n) = isprime(n) && isprime(znorder(Mod(3,n))); \\ Michel Marcus, Oct 28 2016

A309130 Smallest prime factor of A077586(n).

Original entry on oeis.org

7, 127, 2147483647, 170141183460469231731687303715884105727, 47, 338193759479, 231733529, 62914441, 2351, 1399, 295257526626031, 18287, 106937, 863, 4703, 138863, 22590223644617

Offset: 1

Richard N. Smith, Jul 13 2019

A263686 is a subsequence.
Agrees with A263686 in the first four terms, but then the two sequences differ for the first time at n = 5, because prime(5) = 11 is not in A000043.
a(18) = A263686(9) is greater than 1.56*10^17*(2^61-1), see link.
a(n) = A077586(n) iff A077586(n) is prime, A077586(n) is prime for 1 <= n <= 4, but composite for 5 <= n <= 17. The status of A077586(18) = 2^(2^61-1)-1 is unknown. It is conjectured that A077586(n) is composite for all n >= 5.
a(20) = 456959, a(21) = 18384329, a(22) = 198839, a(23) = 2349023, a(24) = A263686(10) is greater than 1.25*10^16*(2^89-1).
Conjecture: All terms are in A122094 (all terms in A263686 are in A122094).
For examples related to that conjecture, see A322568. - Jeppe Stig Nielsen, Aug 29 2019
a(30) = 46559, a(32) = 23671, a(36) = 7151489, a(39) = 4698047, a(41) = 719, a(43) = 1440847, a(45) = 179689, a(47) = 11759383, a(48) = 23602441, a(50) = 9024439, a(51) = 28875361, a(52) = 6301423, a(54) = 2493983, a(56) = 33518137, a(59) = 6727783, a(66) = 95111, a(72) = 1439, a(73) = 99833, a(78) = 38119, a(81) = 26849, a(83) = 8258911, a(86) = 16173559, a(89) = 625343, a(93) = 9743. - Chai Wah Wu, Oct 16 2019

Double Mersennes Prime Search, History.
Eric Weisstein's World of Mathematics, Double Mersenne Number.

Cf. A077586, A263686, A001348, A049479.

A309130(n)=A020639(2^(2^prime(n)-1)-1) \\ For efficiency, use addprimes([large terms of this sequence]). - M. F. Hasler, Mar 01 2025

a(n) = A020639(A077586(n)).

a(n) = A049479(A001348(n)). - M. F. Hasler, Mar 01 2025

A137332 Primes which are equal to the order of 2 modulo a prime q, sorted with respect to the value of q.

Original entry on oeis.org

2, 3, 11, 5, 23, 11, 7, 83, 37, 29, 131, 179, 191, 43, 73, 239, 251, 359, 419, 431, 443, 491, 29, 659, 683, 233, 179, 719, 743, 911, 239, 1019, 1031, 29, 1103, 47, 397, 1223, 79, 461, 1439, 1451, 1499, 1511, 1559, 1583, 557, 113, 431, 577, 601, 1811, 1931

Offset: 1

Joerg Arndt, Apr 07 2008

nonn

This is a multipermutation of the primes A000040 with every prime p appearing exactly A001221(2^p-1) times. - Max Alekseyev, May 01 2008

			The k-th term of the sequence is ord(2 mod A122094(k)).
For example, 223 is the 9th term of A122094 and ord(2 mod 223)=37, so 37 is the 9th term of this sequence.
11 is both the third term because ord(2 mod 23) == 11 and the sixth term because ord(2 mod 89) == 11.
Note both 23 and 89 divide 2^11-1; the third and sixth terms of A122094 are 23 and 89.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..106 from Joerg Arndt)

Cf. A002326, A122094.

Select[MultiplicativeOrder[2, #] & /@ Select[Range[3, 4000, 2], PrimeQ], PrimeQ] (* Amiram Eldar, Apr 04 2020 *)

forprime (p=3, 10^4, r = znorder( Mod(2,p) ); if ( isprime(r), print1(r, ", "); ); );

a(n) = A007733(A122094(n)) = A002326((A122094(n)-1)/2). - Max Alekseyev, May 01 2008

Definition revised by Max Alekseyev, May 01 2008

A213049 Primes p such that the order of 2 mod p is a square.

Original entry on oeis.org

5, 37, 73, 101, 109, 197, 257, 577, 601, 641, 677, 727, 1601, 1801, 2593, 3137, 3389, 3457, 4057, 4357, 5477, 8101, 8837, 10369, 14401, 14407, 16901, 17957, 18253, 18433, 20809, 21317, 22501, 25601, 30977, 33857, 37447, 42437, 44101, 47629, 47653, 50177

Offset: 1

Joerg Arndt, Jun 03 2012

nonn

			The order of 2 mod 601 is 25, which is a square, so 601 is a term.

Vincenzo Librandi, Table of n, a(n) for n = 1..178

Cf. A014662, A014663, A122094.

[NthPrime(n): n in [2..6275] | IsSquare(Modorder(2, NthPrime(n)))]; // Bruno Berselli, Jun 08 2012

{ forprime (p=3, 10^6,
    r = znorder(Mod(2,p));
    if ( issquare(r), print1(p,", ") );
); }

A218582 Primes which do not divide any Mersenne number M(p) = 2^p - 1 with prime p.

Original entry on oeis.org

2, 5, 11, 13, 17, 19, 29, 37, 41, 43, 53, 59, 61, 67, 71, 73, 79, 83, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 173, 179, 181, 191, 193, 197, 199, 211, 227, 229, 239, 241, 251, 257, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337

Offset: 1

Arkadiusz Wesolowski, Nov 03 2012

nonn

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Complement in primes of A122094. Subsequence of A138837.

[p: p in PrimesInInterval(2, 337) | not IsPrime(Modorder(2, p))];

Select[Prime@Range[68], ! PrimeQ@MultiplicativeOrder[2, #] &]

A308079 Pseudoprimes to base 3 that divide a Mersenne number.

Original entry on oeis.org

10974881, 193949641, 717653129, 8762386393, 19683169273, 24802217129, 78618861353, 121271968201, 146050578391, 169905267617, 188684740591, 232153956569, 290762221753, 306091598201, 336675266287, 394233108121, 592050558553

Offset: 1

Jeppe Stig Nielsen, May 11 2019

nonn

Members of A005935 that divide a member of A001348.
Odd members k of A005935 such that the multiplicative order of 2 modulo k is a prime. Odd members k of A005935 such that A002326((k-1)/2) is prime.
The known entries are proper divisors of a Mersenne number. It is not known if the Mersenne number itself can belong to this sequence.

			10974881 is in the sequence because it divides 2^239 - 1 (and 239 is prime), it is not a prime, but 3^10974880 === 1 (mod 10974881).

Amiram Eldar, Table of n, a(n) for n = 1..202 (terms below 10^15)
Mersenne Forum, Composite PRP (discussion thread).

Intersection of A005935 and A122094.
Subsequence of A052155.
Cf. A001348, A002326.

forstep(n=3,+oo,2,Mod(3,n)^(n-1)==1&&!ispseudoprime(n)&&ispseudoprime(znorder(Mod(2,n)))&&print1(n,", "))

A379161 Primes p such that the multiplicative order of 7 modulo p is prime.

Original entry on oeis.org

19, 29, 47, 59, 83, 167, 223, 227, 311, 367, 383, 389, 419, 439, 467, 479, 503, 563, 587, 607, 653, 719, 809, 839, 887, 971, 983, 1123, 1307, 1319, 1447, 1487, 1543, 1733, 1811, 1823, 1907, 1997, 2063, 2099, 2153, 2239, 2383, 2579, 2741, 2801, 2819, 2837, 2887, 2903, 2909, 2999, 3023, 3083, 3167, 3463, 3547

Offset: 1

Vincenzo Librandi, Dec 17 2024

nonn

Odd primes that divide 7^p-1 for some prime p [after Robert Israel].

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A122094, A277048, A277049.

[p: p in PrimesInInterval(2, 4000) | IsPrime(Modorder(7, p))];

filter:= n -> isprime(n) and isprime(numtheory:-order(7,n)):
select(filter, [2,3,5, seq(i,i=11..10000,2)]); # Robert Israel, Jan 03 2025

Select[Prime@Range@4000,PrimeQ@MultiplicativeOrder[7,#]&]

cp's OEIS Frontend

A322568 Integers k such that the least prime factor of 2^k - 1 is not in A122094.

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A089162 Triangle read by rows formed by the prime factors of Mersenne number 2^prime(n) - 1, n >= 1.

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A263686 Smallest prime factor of double Mersenne numbers.

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A277048 Primes p such that the multiplicative order of 3 modulo p is prime.

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A309130 Smallest prime factor of A077586(n).

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A137332 Primes which are equal to the order of 2 modulo a prime q, sorted with respect to the value of q.

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A213049 Primes p such that the order of 2 mod p is a square.

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Magma

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A218582 Primes which do not divide any Mersenne number M(p) = 2^p - 1 with prime p.

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