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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: A340320 A059998 A339731 * A339082 A329907 A329958 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 May 8 19:28 EDT 2021. Contains 343666 sequences. (Running on oeis4.)