

A234475


Number of ways to write n = k + m with 2 < k <= m such that q(phi(k)*phi(m)/4) + 1 is prime, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).


14



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

1,8


COMMENTS

Conjecture: a(n) > 0 for all n > 5.
This implies that there are infinitely many primes p with p  1 a term of A000009.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..2525
Z.W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014


EXAMPLE

a(6) = 1 since 6 = 3 + 3 with q(phi(3)*phi(3)/4) + 1 = q(1) + 1 = 2 prime.
a(76) = 1 since 76 = 18 + 58 with q(phi(18)*phi(58)/4) + 1 = q(42) + 1 = 1427 prime.
a(197) = 1 since 197 = 4 + 193 with q(phi(4)*phi(193)/4) + 1 = q(96) + 1 = 317789.
a(356) = 1 since 356 = 88 + 268 with q(phi(88)*phi(268)/4) + 1 = q(1320) + 1 = 35940172290335689735986241 prime.


MATHEMATICA

f[n_, k_]:=PartitionsQ[EulerPhi[k]*EulerPhi[nk]/4]+1
a[n_]:=Sum[If[PrimeQ[f[n, k]], 1, 0], {k, 3, n/2}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A000009, A000010, A000040, A232504, A233307, A233346, A233547, A233390, A233393, A234309, A234310, A234337, A234344, A234347, A234359, A234360, A234361, A234451, A234470, A234514, A234530, A234567, A234569
Sequence in context: A083447 A252489 A059998 * A309969 A036041 A252759
Adjacent sequences: A234472 A234473 A234474 * A234476 A234477 A234478


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Dec 26 2013


STATUS

approved



