A281317 Primes p such that p == i mod d(i) where d(i) are the prime divisors of 2p+1.

7, 13, 37, 67, 157, 337, 367, 607, 787, 937, 1093, 3037, 3307, 7717, 9187, 12757, 15187, 19687, 27337, 35437, 42187, 49207, 69457, 75937, 267907, 347287, 683437, 744187, 797161, 882367, 1148437, 1458607, 1736437, 2067187, 2870437, 2929687, 3125587, 4823437

Offset: 1

Author

Michel Lagneau, Jan 20 2017

Keywords

• nonn

Comments

Subsequence of A053176.

a(n)== 1 mod 6 or a(n)== 1, 7 mod 12. A majority of members of the sequence are congruent to 7 mod 10.

omega(2*a(n)+1) = 1 for n = 2, 11, 29,... => 2*a(n)+1 = 3^3, 3^7, 3^13,... where omega(n) = A001221(n).

Examples

			157 is in the sequence because  2*157 + 1 = 315 = 3 ^ 2 * 5 * 7 => 157 == 1 (mod 3), 157 == 2 (mod 5) and 157 == 3 (mod 7).
		

Crossrefs

Cf. A001221, A053176, A053177.

Programs

Maple

with(numtheory):
for n from 2 to 10^5 do:
  p:=ithprime(n):q:=2*p+1:x:=factorset(q):n1:=nops(x):j:=0:
   for i from 1 to n1 do:
     if irem(p,x[i])=i
      then j:=j+1:
      else
     fi:
   od:
    if j=n1
     then
     printf(`%d, `,p):
     else
    fi:
  od:

Mathematica

Select[Prime@ Range[10^6], Function[p, Function[i, Times @@ Boole@ MapIndexed[Mod[p, #1] == First@ #2 &, FactorInteger[i][[All, 1]]] > 0][2 p + 1]]] (* Michael De Vlieger, Jan 20 2017 *)