A233549 Number of ways to write n = p + q (q > 0) with p and (phi(p)*phi(q))^4 + 1 prime, where phi(.) is Euler's totient function (A000010).

0, 0, 1, 2, 2, 3, 3, 2, 3, 2, 1, 3, 1, 4, 3, 3, 4, 4, 6, 1, 1, 1, 4, 1, 2, 2, 4, 4, 1, 6, 7, 3, 4, 3, 4, 3, 3, 5, 2, 3, 5, 3, 1, 3, 5, 3, 3, 5, 6, 4, 4, 5, 4, 3, 4, 6, 4, 4, 3, 4, 5, 4, 2, 2, 4, 3, 6, 1, 4, 2, 8, 9, 2, 5, 5, 4, 2, 3, 4, 3, 6, 1, 7, 5, 8, 5, 4, 4, 4, 10, 10, 6, 4, 8, 4, 3, 4, 6, 6, 2

Offset

1,4

Comments

Conjecture: (i) a(n) > 0 for all n > 2.

(ii) If n > 2 is not equal to 26, then there is a prime p < n with (phi(p)*phi(n-p))^2 + 1 prime.

(iii) If n > 3 is different from 9 and 16, then there is a prime p < n with ((p+1)*phi(n-p))^2 + 1 prime.

Part (i) of the conjecture implies that there are infinitely many primes of the form x^4 + 1. We have verified it for n up to 10^7.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(11) = 1 since 11 = 2 + 9 with 2 and (phi(2)*phi(9))^4 + 1 = 6^4 + 1 = 1297 both prime.
a(13) = 1 since 13 = 5 + 8 with 5 and (phi(5)*phi(8))^4 + 1 = 16^4 + 1 = 65537 both prime.
a(258) = 1 since 258 = 167 + 91 with 167 and (phi(167)*phi(91))^4 + 1 = (166*72)^4 + 1 = 20406209352892417 both prime.

Mathematica

a[n_]:=Sum[If[PrimeQ[((Prime[k]-1)*EulerPhi[n-Prime[k]])^4+1], 1, 0], {k, 1, PrimePi[n-1]}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000010, A000040, A000068, A037896, A233542, A233544, A233547.

Sequence in context: A055093 A196058 A081844 * A334475 A110012 A233542

Adjacent sequences: A233546 A233547 A233548 * A233550 A233551 A233552

Keyword

nonn

Author

Zhi-Wei Sun, Dec 12 2013

Status

approved