A235805 a(n) = |{0 < k < n - 2: 2*m + 1, m*(m+1) - prime(m) and m*(m+1) - prime(m+1) are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.

0, 0, 0, 0, 0, 2, 3, 2, 4, 4, 4, 4, 4, 4, 2, 6, 3, 7, 4, 2, 7, 3, 5, 4, 4, 6, 6, 6, 4, 4, 7, 8, 9, 6, 6, 11, 8, 10, 6, 6, 12, 8, 13, 6, 12, 8, 13, 10, 7, 14, 10, 11, 11, 11, 16, 14, 13, 9, 15, 11, 23, 14, 12, 11, 12, 10, 14, 8, 15, 9, 14, 13, 11, 12, 9, 19, 9, 14, 11, 16, 8, 14, 5, 13, 8, 13, 9, 13, 10, 15, 10, 11, 12, 17, 8, 13, 10, 11, 7, 18

Offset

1,6

Comments

Conjecture: a(n) > 0 for all n > 5.

This implies that there are infinitely many odd primes p = 2*m + 1 with m*(m+1) - prime(m) and m*(m+1)- prime(m+1) both prime.

Links

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

Example

a(8) = 2 since phi(4) + phi(4)/2 = 3 with 2*3 + 1 = 7, 3*4 - prime(3) = 7 and 3*4 - prime(4) = 5 all prime, and phi(5) + phi(3)/2 = 5 with 2*5 + 1 = 11, 5*6 - prime(5) = 19 and 5*6 - prime(6) = 17 all prime.

Mathematica

q[n_]:=PrimeQ[2n+1]&&PrimeQ[n(n+1)-Prime[n]]&&PrimeQ[n(n+1)-Prime[n+1]]
f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/2
a[n_]:=Sum[If[q[f[n, k]], 1, 0], {k, 1, n-3}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000010, A000040, A235592, A235728, A235806.

Sequence in context: A304739 A304735 A318048 * A168231 A235728 A205546

Adjacent sequences: A235802 A235803 A235804 * A235806 A235807 A235808

Keyword

nonn

Author

Zhi-Wei Sun, Jan 16 2014

Status

approved