A277201 -id:A277201 - cp's OEIS frontend

A176116 Primes p such that p^4+1 = 2q where q is prime.

3, 5, 7, 11, 13, 17, 23, 29, 61, 71, 73, 103, 113, 199, 251, 313, 317, 337, 353, 419, 431, 449, 463, 479, 487, 503, 523, 607, 613, 643, 677, 701, 719, 761, 769, 811, 821, 829, 857, 883, 919, 997, 1013, 1019, 1049, 1087, 1123, 1163, 1259, 1327, 1381, 1483, 1493

Offset: 1

Kevin Batista (kevin762401(AT)yahoo.com), Apr 08 2010

			3^4+1 = 2*41; 5^4+1 = 2*313; 7^4+1 = 2*1201; 11^4+1 = 2*7321.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Subsequence of A096169.

Cf. A277201 (resulting primes).

Select[Prime[Range[250]],PrimeQ[(#^4+1)/2]&] (* Harvey P. Dale, Jul 20 2012 *)

lista(nn) = forprime(p=3, nn, if (isprime((p^4+1)/2), print1(p, ", "));); \\ Michel Marcus, Oct 03 2016

Edited by Ray Chandler, Apr 10 2010

A382872 For n >= 1, a(n) is the number of divisors of the Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n) (A018804).

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 2, 6, 4, 4, 4, 8, 3, 4, 6, 10, 4, 6, 2, 12, 4, 6, 6, 9, 4, 6, 5, 8, 4, 8, 2, 10, 8, 6, 6, 16, 2, 4, 4, 18, 5, 8, 4, 16, 8, 8, 4, 20, 4, 8, 8, 12, 8, 6, 8, 12, 4, 6, 6, 24, 3, 4, 8, 9, 9, 12, 4, 16, 9, 8, 4, 24, 4, 4, 6, 8, 8, 8, 2, 20

Offset: 1

Ctibor O. Zizka, Apr 07 2025

nonn

a(n) is from A005408 for n from {1, 5, 13, 24, 27, 41, 61, 64, 65, 69, 99, 113, ...}.

a(n) is from A065091 for n from {5, 13, 27, 41, 61, 135, 181, 205, 313, 421, ...}.

			For n = 5, a(5) = A000005(A018804(5)) = A000005(9) = 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A005382, A005408, A018804, A065091, A067756, A277201.

f:= proc(n) local i; numtheory:-tau(add(igcd(i,n),i=1..n)) end proc:
map(f, [$1..100]); # Robert Israel, May 07 2025

f[p_, e_] := (e*(p - 1)/p + 1)*p^e; a[n_] := DivisorSigma[0, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Apr 07 2025 *)

a(n) = numdiv(sumdiv(n, d, n*eulerphi(d)/d)); \\ Michel Marcus, Apr 07 2025

a(n) = A000005(A018804(n)).
a(A005382(n)) = 2.
a(A067756(n)) = 3.
a(A277201(n)) = 5.

cp's OEIS Frontend

A176116 Primes p such that p^4+1 = 2q where q is prime.

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