A234567 Number of ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 + 1 and P(p-1) are both prime, where phi(.) is Euler's totient function and P(.) is the partition function (A000041).

0, 0, 0, 1, 2, 1, 1, 3, 2, 2, 3, 2, 4, 2, 4, 4, 2, 4, 3, 5, 1, 3, 2, 3, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 2, 0, 1, 2, 1, 1, 2, 1, 2, 3, 2, 8, 2, 1, 2, 2, 3, 3, 1, 2, 7, 0, 2, 3, 3, 4, 5, 7, 3, 4, 1, 9, 1, 4, 3, 1, 2

Offset

1,5

Comments

Conjecture: (i) a(n) > 0 for all n > 727.

(ii) For the strict partition function q(.) (cf. A000009), any n > 93 can be written as k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 + 1 and q(p-1) - 1 are both prime.

(iii) If n > 75 is not equal to 391, then n can be written as k + m with k > 0 and m > 0 such that f(k,m) - 1, f(k,m) + 1 and q(f(k,m)) + 1 are all prime, where f(k,m) = phi(k) + phi(m)/2.

Part (i) of the conjecture implies that there are infinitely many primes p with P(p-1) prime.

Links

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

Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Example

a(21) = 1 since 21 = 6 + 15 with  phi(6) + phi(15)/2 + 1 = 7 and P(6) = 11 both prime.
a(700) = 1 since 700 = 247 + 453 with phi(247) + phi(453)/2 + 1 = 367 and P(366) = 790738119649411319 both prime.
a(945) = 1 since 945 = 687 + 258 with phi(687) + phi(258)/2 + 1 = 499 and P(498) = 2058791472042884901563 both prime.

Mathematica

f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/2
q[n_, k_]:=PrimeQ[f[n, k]+1]&&PrimeQ[PartitionsP[f[n, k]]]
a[n_]:=Sum[If[q[n, k], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000009, A000010, A000040, A000041, A234470, A234475, A234514, A234530.

Sequence in context: A060118 A329143 A219032 * A241950 A316776 A029308

Adjacent sequences: A234564 A234565 A234566 * A234568 A234569 A234570

Keyword

nonn

Author

Zhi-Wei Sun, Dec 28 2013

Status

approved