A280003 -id:A280003 - cp's OEIS frontend

A280004 Numbers k such that 9*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

Original entry on oeis.org

67, 9431, 461081, 2543551

Offset: 1

Arkadiusz Wesolowski, Feb 21 2017

Fernando (Remark 5.2) shows that all terms are odd. - Jeppe Stig Nielsen, Jan 02 2025

R. Fernando, Morehead-like restrictions on Fermat divisors.
Proth Search Page, Prime factors of Fermat numbers
Eric Weisstein's World of Mathematics, Fermat Number

Subsequence of A002256.

Cf. A000215, A050528, A057778, A201364, A204620, A226366, A280003.

A298669 Let b(n) = 2^n with n >= 2, and let c = k*b(n) + 1 for k >= 1; then a(n) is the smallest k such that c is prime and such that A007814(r(n)) = A007814(k) + n where r(n) is the remainder of 2^(b(n)/4) mod c, or 0 if no such k exists.

Original entry on oeis.org

0, 0, 1, 8, 1024, 5, 1071, 6443, 52743, 1184, 11131, 39, 7, 856079, 3363658, 9264, 3150, 1313151, 13, 33629, 555296667, 534689, 8388607, 5, 512212693, 193652, 286330, 282030, 7224372579, 1120049, 149041

Offset: 2

Arkadiusz Wesolowski, Jan 24 2018

nonn

a(n-2) <= A007117(n).

a(33) <= 5463561471303.

Wilfrid Keller, Fermat factoring status

Cf. A007117, A007814, A298670.

print1(0, ", "0", "); for(n=4, 32, b=2^n; k=1; t=0; while(t<1, c=k*b+1; if(isprime(c), r=Mod(2, c)^(b/4); if(lift(r/b)<=k, if(valuation(lift(r), 2)==valuation(k, 2)+n, t=1; print1(k, ", ")))); k++));

For n >= 1, a(A204620(n)) = 3; a(A226366(n)) = 5; a(A280003(n)) = 7.

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, ", ")));

cp's OEIS Frontend

A280004 Numbers k such that 9*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

Views

Author

Keywords

Comments

Links

Crossrefs

A298669 Let b(n) = 2^n with n >= 2, and let c = k*b(n) + 1 for k >= 1; then a(n) is the smallest k such that c is prime and such that A007814(r(n)) = A007814(k) + n where r(n) is the remainder of 2^(b(n)/4) mod c, or 0 if no such k exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

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