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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).

%I A351170 #27 Mar 17 2022 23:57:22
%S A351170 22,54,454,1254,6054,31254,84054,296454,432454,806454,832054,1022454,
%T A351170 2398054,2622054,2761254,3100054,3251254,3458454,3781254,4898454,
%U A351170 5216454,5611254,5678454,7722454,8446054,8694454,8778054,11568054,12054054,12852454,14204454,16074454
%N 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).
%H A351170 Michel Lagneau, <a href="/A351170/a351170.pdf">a(n)==54 (mod 100)</a>
%F A351170 For n>1, a(n) == 54 (mod 100) (see proof above).
%F A351170 a(n) = 2*(A096012(n)+1)^2+4 = 2*A108814(n)^2+4. - _Alois P. Heinz_, Feb 04 2022
%F A351170 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
%e A351170 a(3) = 454 because A096012(3) = 14, 14^2+1 = 197, (14+2)^2+1 = 257, and 197 + 257 = 454.
%p A351170 nn:=3000:
%p A351170 for n from 2 by 2 to nn do:
%p A351170   p1:=n^2+1:p2:=(n+2)^2+1:
%p A351170    if isprime(p1) and isprime(p2)
%p A351170     then
%p A351170     s:=p1+p2:printf(`%d, `,s):
%p A351170     else
%p A351170    fi:
%p A351170 od:
%t A351170 f[n_] := 2*n^2 + 4*n + 6; f /@ Select[Range[3000], And @@ PrimeQ[{#^2 + 1, (# + 2)^2 + 1}] &] (* _Amiram Eldar_, Feb 04 2022 *)
%o A351170 (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
%Y A351170 Cf. A002496, A005574, A096012, A108814, A206328.
%K A351170 nonn
%O A351170 1,1
%A A351170 _Michel Lagneau_, Feb 04 2022