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 A234615 Number of ways to write n = k + m with k > 0 and m > 0 such that p = prime(k) + phi(m) and q(p) - 1 are both prime, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009). 12
 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 4, 3, 5, 4, 2, 6, 6, 6, 5, 4, 5, 6, 4, 6, 5, 5, 2, 4, 5, 6, 5, 7, 4, 6, 6, 8, 3, 3, 6, 7, 7, 4, 4, 4, 4, 7, 7, 3, 3, 4, 4, 6, 5, 4, 5, 5, 7, 1, 3, 4, 7, 5, 5, 6, 3, 7, 11, 5, 4, 5, 4, 7, 6, 4, 2, 7, 9, 7, 5, 5, 6, 5, 10, 7, 4, 3, 4, 6, 3, 4, 9, 5, 3, 5, 6, 5, 3, 6, 2, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS Conjecture: (i) a(n) > 0 for all n > 7. (ii) Any integer n > 7 not equal to 15 can be written as k + m with k > 0 and m > 0 such that p = prime(k) + phi(m) and q(p) + 1 are both prime. (iii) Any integer n > 83 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m)/2 is a square. Also, each integer n > 45 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m)/2 is a triangular number. Clearly, part (i) of this conjecture implies that there are infinitely many primes p with q(p) - 1 also prime (cf. A234644). LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..8000 Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014 EXAMPLE a(6) = 1 since 6 = 2 + 4 with prime(2) + phi(4) = 5 and q(5) - 1 = 2 both prime. a(58) = 1 since 58 = 12 + 46 with prime(12) + phi(46) = 59 and q(59) - 1 = 9791 both prime. a(526) = 1 since 526 = 389 + 137 with prime(389) + phi(137) = 2819 and q(2819) - 1 = 326033386646595458662191828888146112979 both prime. MATHEMATICA f[n_, k_]:=Prime[k]+EulerPhi[n-k] q[n_, k_]:=PrimeQ[f[n, k]]&&PrimeQ[PartitionsQ[f[n, k]]-1] a[n_]:=Sum[If[q[n, k], 1, 0], {k, 1, n-1}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000009, A000010, A000040, A234475, A234514, A234530, A234567, A234569, A234572, A234644 Sequence in context: A282561 A237598 A138241 * A029145 A238999 A097986 Adjacent sequences:  A234612 A234613 A234614 * A234616 A234617 A234618 KEYWORD nonn AUTHOR Zhi-Wei Sun, Dec 28 2013 STATUS approved

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Last modified May 25 03:14 EDT 2019. Contains 323539 sequences. (Running on oeis4.)