A235682 Number of ways to write n = k + m with k > 0 and m > 2 such that p = phi(k) + phi(m)/2 + 1, prime(p) - p + 1 and p*(p+1) - prime(p) are all prime, where phi(.) is Euler's totient function.

0, 0, 0, 1, 2, 1, 1, 3, 2, 1, 3, 1, 1, 2, 2, 3, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 3, 0, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 6, 3, 6, 0, 6, 4, 5, 3, 1, 3, 4, 2, 3, 4, 1, 8, 6, 4, 8, 8

Offset

1,5

Comments

Conjecture: a(n) > 0 for all n > 84.

Clearly, this implies that there are infinitely many primes p with prime(p) - p + 1 and p*(p+1) - prime(p) both prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(10) = 1 since 10 = 1 + 9 with phi(1) + phi(9)/2 + 1 = 5, prime(5) - 5 + 1 = 7 and 5*6 - prime(5) = 19 all prime.
a(95) = 1 since 95 = 62 + 33 with phi(62) + phi(33)/2 + 1 = 41, prime(41) - 41 + 1 = 139 and 41*42 - prime(41) = 1543 all prime.
a(421) = 1 since 421 = 289 + 132 with phi(289) + phi(132)/2 + 1 = 293, prime(293) - 293 + 1 = 1621 and 293*294 - prime(293) = 84229 all prime.

Mathematica

PQ[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n+1]&&PrimeQ[n(n+1)-Prime[n]]
f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/2+1
a[n_]:=Sum[If[PQ[f[n, k]], 1, 0], {k, 1, n-3}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000010, A000040, A232353, A234694, A234695, A235189, A235330, A235613, A235614, A235661, A235681.

Sequence in context: A152538 A345033 A329359 * A324830 A141110 A325758

Adjacent sequences: A235679 A235680 A235681 * A235683 A235684 A235685

Keyword

nonn

Author

Zhi-Wei Sun, Jan 13 2014

Status

approved