A351170 Consider the primes of the form p(m)=m^2+1 such that p(m+2) is also prime for some m. The sequence lists the sums p(m) + p(m+2).

22, 54, 454, 1254, 6054, 31254, 84054, 296454, 432454, 806454, 832054, 1022454, 2398054, 2622054, 2761254, 3100054, 3251254, 3458454, 3781254, 4898454, 5216454, 5611254, 5678454, 7722454, 8446054, 8694454, 8778054, 11568054, 12054054, 12852454, 14204454, 16074454

Offset: 1

Author

Michel Lagneau, Feb 04 2022

Keywords

• nonn

Examples

			a(3) = 454 because A096012(3) = 14, 14^2+1 = 197, (14+2)^2+1 = 257, and 197 + 257 = 454.
		

Links

• Michel Lagneau, a(n)==54 (mod 100)

Crossrefs

Cf. A002496, A005574, A096012, A108814, A206328.

Programs

Maple

nn:=3000:
for n from 2 by 2 to nn do:
  p1:=n^2+1:p2:=(n+2)^2+1:
   if isprime(p1) and isprime(p2)
    then
    s:=p1+p2:printf(`%d, `,s):
    else
   fi:
od:

Mathematica

f[n_] := 2*n^2 + 4*n + 6; f /@ Select[Range[3000], And @@ PrimeQ[{#^2 + 1, (# + 2)^2 + 1}] &] (* Amiram Eldar, Feb 04 2022 *)

PARI

lista(nn) = {for (m=1, nn, if (isprime(m^2+1) && isprime(m^2+4*m+5), print1(2*m^2+4*m+6, ", ")););} \\ Michel Marcus, Feb 04 2022

Formula

For n>1, a(n) == 54 (mod 100) (see proof above).

a(n) = 2*(A096012(n)+1)^2+4 = 2*A108814(n)^2+4. - Alois P. Heinz, Feb 04 2022

For n > 1, a(n) mod 400 = 54; a(n) mod 1200 = 54 or 454; a(n) mod 2000 = 54, 454, or 1254; a(n) mod 54, 454, 1254, or 2454. - Jon E. Schoenfield, Feb 04 2022