A232353 - OEIS

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A232353

Number of ways to write n = k + m with k > 0 and m > 0 such that p = prime(k) + phi(m) and p*(p+1) - prime(p) are both prime, where phi(.) is Euler's totient function.

0, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 2, 1, 1, 2, 4, 3, 2, 6, 3, 4, 4, 5, 5, 3, 4, 2, 6, 5, 4, 6, 5, 4, 6, 7, 1, 6, 4, 8, 6, 6, 7, 4, 5, 10, 5, 3, 4, 6, 7, 6, 6, 9, 6, 3, 7, 7, 10, 5, 9, 7, 7, 6, 5, 8, 9, 4, 6, 9, 8, 5, 8, 5, 8, 8, 5, 6, 7, 9, 10, 8, 8, 8, 11, 10, 11, 7, 8, 13, 9, 6, 12, 10, 5, 9, 7, 8, 14, 8

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OFFSET

1,10

COMMENTS

Conjecture: a(n) > 0 for all n > 7.

This implies that there are infinitely many primes p with p*(p+1) - prime(p) prime.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

EXAMPLE

a(14) = 1 since 14 = 4 + 10 with prime(4) + phi(10) = 11 and 11*12 - prime(11) = 101 both prime.

a(15) = 1 since 15 = 6 + 9 with prime(6) + phi(9) = 19 and 19*20 - prime(19) = 313 both prime.

a(37) = 1 since 37 = 23 + 14 with prime(23) + phi(14) = 89 and 89*90 - prime(89) = 7549 both prime.

MATHEMATICA

PQ[n_]:=PrimeQ[n]&&PrimeQ[n(n+1)-Prime[n]]

f[n_, k_]:=Prime[k]+EulerPhi[n-k]

a[n_]:=Sum[If[PQ[f[n, k]], 1, 0], {k, 1, n-1}]

Table[a[n], {n, 1, 100}]

CROSSREFS

Cf. A000010, A000040, A234694, A235592, A235613, A235614, A235661.

Sequence in context: A293431 A078573 A143786 * A366987 A304093 A238580

Adjacent sequences: A232350 A232351 A232352 * A232354 A232355 A232356

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 13 2014

STATUS

approved