

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
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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

ZhiWei 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[nk]
q[n_, k_]:=PrimeQ[f[n, k]]&&PrimeQ[PartitionsQ[f[n, k]]1]
a[n_]:=Sum[If[q[n, k], 1, 0], {k, 1, n1}]
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

ZhiWei Sun, Dec 28 2013


STATUS

approved



