A057023 -id:A057023 - cp's OEIS frontend

A361180 Primes p such that the odd part of p - 1 is upper-bounded by the dyadic valuation of p - 1.

3, 5, 17, 97, 193, 257, 641, 769, 12289, 18433, 40961, 65537, 114689, 147457, 163841, 786433, 1179649, 5767169, 7340033, 13631489, 23068673, 167772161, 469762049, 2013265921, 2281701377, 3221225473, 3489660929, 12348030977, 77309411329, 206158430209, 2061584302081, 2748779069441

Offset: 1

Lorenzo Sauras Altuzarra, Mar 03 2023

nonn

Primes of the form k*2^m + 1 where k <= m and k is odd. - David A. Corneth, Mar 03 2023

Primes prime(k) such that A057023(k) <= A023506(k). - Michel Marcus, Mar 09 2023

			3 is a term because the odd part of 2 is 1, the dyadic valuation of 2 is 1 and 1 <= 1.
641 = 5*2^7 + 1 is a term because the odd part of 640 is 5, the dyadic valuation of 640 is 7 and 5 <= 7.

Cf. A000040 (primes), A000265 (odd part), A007814 (dyadic valuation).

Cf. A057023, A023506.

# Maple program due to David A. Corneth, Mar 03 2023
aList := proc(upto)
   local i, j, p, R:
   R := {}:
   for i from 1 to ilog2(upto) do
      for j from 1 to min(i, floor(upto/2^i)) do
         p := j*2^i+1:
         if isprime(p) then R := `union`(R, {p}): fi: od: od:
   R: end:
aList(10^12);

isok(p) = if (isprime(p), my(m=valuation(p-1,2)); (p-1)/2^m <= m); \\ Michel Marcus, Mar 03 2023

upto(n) = {my(res = List()); for(i = 1, logint(n, 2), forstep(j = 1, min(i, n>>i), 2, if(isprime((j<David A. Corneth, Mar 03 2023

a(17)..a(27) from Michel Marcus, Mar 03 2023

More terms from David A. Corneth, Mar 03 2023

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

A361180 Primes p such that the odd part of p - 1 is upper-bounded by the dyadic valuation of p - 1.

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

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