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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 (list; graph; refs; listen; history; text; internal format)
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

Zhi-Wei 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[n-k]/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: A252489 A332250 A059998 * A329907 A329958 A309969

Adjacent sequences:  A234472 A234473 A234474 * A234476 A234477 A234478

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Dec 26 2013

STATUS

approved

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Last modified July 6 06:05 EDT 2020. Contains 335476 sequences. (Running on oeis4.)