

A236456


Number of ordered ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(nk)/4  1, q = p + 2 and r = prime(q) + 2 are all prime, where phi(.) is Euler's totient function.


9



0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 2, 3, 3, 2, 4, 2, 2, 4, 5, 4, 2, 3, 1, 3, 2, 3, 4, 3, 4, 5, 0, 2, 2, 3, 2, 4, 2, 4, 3, 2, 2, 1, 2, 5, 2, 3, 1, 4, 2, 2, 4, 1, 4, 1, 5, 4, 2, 2, 1, 2, 1, 5, 3, 3, 1, 2, 2, 4, 1, 3, 4, 2, 2, 1, 0, 2, 4, 2, 1, 3, 1, 4, 3, 5, 3, 2, 1, 3, 2, 3, 2, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,19


COMMENTS

Conjecture: a(n) > 0 for all n > 357.
This is much stronger than the twin prime conjecture. Actually it implies that there are infinitely many primes p such that {p, p + 2} and {prime(p+2), prime(p+2) + 2} are both twin prime pairs. See A236457 for such primes p.


LINKS



EXAMPLE

a(18) = 1 since 18 = 3 + 15 with phi(3) + phi(15)/4  1 = 3, 3 + 2 = 5 and prime(5) + 2 = 13 all prime.
a(50) = 1 since 50 = 16 + 34 with phi(16) + phi(34)/4  1 = 11, 11 + 2 = 13 and prime(13) + 2 = 43 all prime.
a(929) = 1 since 929 = 441 + 488 with phi(441) + phi(488)/4  1 = 252 + 60  1 = 311, 311 + 2 = 313 and prime(313) + 2 = 2083 all prime.


MATHEMATICA

p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[Prime[n+2]+2]
f[n_, k_]:=EulerPhi[k]+EulerPhi[nk]/41
a[n_]:=Sum[If[p[f[n, k]], 1, 0], {k, 1, n1}]
Table[a[n], {n, 1, 100}]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



