A023394 -id:A023394 - cp's OEIS frontend

A283051 Positive integers n such that none of the primes of the form k*2^n + 1 (with k odd) divide any Fermat number F(m) = 2^(2^m) + 1, m >= 0.

Original entry on oeis.org

3, 5, 6, 10

Offset: 1

Arkadiusz Wesolowski, Feb 27 2017

Conjecture: sequence is infinite.

a(5) >= 18.

Proth Search Page, Prime factors of Fermat numbers
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A019434, A023394, A228845, A228846, A229857, A231203.

A342974 Primes p such that the order of 2 modulo p is not divisible by the largest odd divisor of p - 1.

Original entry on oeis.org

31, 43, 109, 127, 151, 157, 223, 229, 241, 251, 277, 283, 307, 331, 397, 431, 433, 439, 457, 499, 571, 601, 631, 641, 643, 673, 683, 691, 727, 733, 739, 811, 911, 919, 953, 971, 997, 1013, 1021, 1051, 1069, 1093, 1103, 1163, 1181, 1321, 1327, 1399, 1423, 1429

Offset: 1

Arkadiusz Wesolowski, Apr 01 2021

nonn

Every prime factor of a composite Fermat number belongs to this sequence.
If a prime of the form 3*2^k + 1 belongs to this sequence, then k is in A204620 (see Golomb).
Primes p such that A014664(primepi(p)) is not divisible by A057023(primepi(p)). - Michel Marcus, Apr 26 2021

Solomon W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.

Cf. A014664, A023394, A057023, A204620, A226366, A280003, A280004.

Select[Prime@Range@300,Mod[MultiplicativeOrder[2,#],Max@Select[Divisors[#-1],OddQ]]!=0&] (* Giorgos Kalogeropoulos, Apr 02 2021 *)

forprime(p=3, 1429, if(Mod(znorder(Mod(2, p)), (p-1)>>valuation(p-1, 2)), print1(p, ", ")));

A354183 Primes p such that p divides 2^((p-1)/x) - 1, where x is the greatest prime factor of p - 1.

Original entry on oeis.org

17, 109, 151, 241, 251, 257, 331, 433, 631, 641, 673, 683, 1321, 1429, 1459, 1613, 2917, 3191, 3457, 3889, 4733, 4861, 5153, 5419, 6337, 7001, 7351, 8581, 9719, 11119, 11251, 11471, 12101, 13367, 13553, 13669, 14323, 14449, 15121, 17539, 18503, 20231, 20857

Offset: 1

Arkadiusz Wesolowski, May 18 2022

nonn

Together with 3 and 5, supersequence of A023394.

Are there any odd integers k (k is not a Sierpiński number) such that every prime of the form k*2^m + 1 (m >= 1) does not belong to the sequence?

Cf. A023394.

gpf:=func; [p: p in PrimesUpTo(20857) | Modexp(2, Truncate((p-1)/gpf(p-1)), p) eq 1];

Select[Prime[Range[2500]], PowerMod[2, (# - 1)/FactorInteger[# - 1][[-1, 1]], #] == 1 &] (* Amiram Eldar, May 19 2022 *)

isok(p) = if (isprime(p) && (p>2), my(x=vecmax(factor(p-1)[,1])); ((2^((p-1)/x) - 1) % p) == 0); \\ Michel Marcus, May 19 2022

A372867 Distinct terms in A242017, listed in the order of their appearance.

Original entry on oeis.org

3, 5, 17, 97, 641, 257, 193, 274177, 65537, 449, 59649589127497217, 769, 1238926361552897, 5441, 5953, 2424833, 7873, 2753, 3329, 10753, 45592577, 18433, 4673, 15937, 444929, 11777, 12161, 21698561, 6977

Offset: 1

Jean-Marc Rebert, May 15 2024

Conjecture: every term except 3 belongs to A366609. - Bill McEachen, Jun 12 2024

Cf. A023394, A070592, A093179, A242017.

cp's OEIS Frontend

A283051 Positive integers n such that none of the primes of the form k*2^n + 1 (with k odd) divide any Fermat number F(m) = 2^(2^m) + 1, m >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

A342974 Primes p such that the order of 2 modulo p is not divisible by the largest odd divisor of p - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A354183 Primes p such that p divides 2^((p-1)/x) - 1, where x is the greatest prime factor of p - 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

PARI

A372867 Distinct terms in A242017, listed in the order of their appearance.

Views

Author

Keywords

Comments

Crossrefs