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A235358 a(n) = |{0 < k < n: g(n,k) - 1, g(n,k) + 1 and q(g(n,k)) - 1 are all prime with g(n,k) = phi(k) + phi(n-k)/8}|, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009). 4
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 3, 1, 2, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 2, 0, 2, 1, 2, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,42

COMMENTS

Conjecture: a(n) > 0 for all n > 1234.

See also part (ii) of the conjecture in A235343.

We have verified the conjecture for n up to 100000.

LINKS

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

EXAMPLE

a(50) = 1 since phi(10) + phi(40)/4 = 6 with 6 - 1, 6 + 1 and q(6) - 1 = 3 all prime.

MATHEMATICA

f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/8

p[n_, k_]:=PrimeQ[f[n, k]-1]&&PrimeQ[f[n, k]+1]&&PrimeQ[PartitionsQ[f[n, k]]-1]

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

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

CROSSREFS

Cf. A000009, A000010, A000040, A001359, A006512, A014574, A234514, A234567, A234615, A235343, A235344, A235346, A235356, A235357.

Sequence in context: A056931 A139569 A201590 * A086249 A176784 A176511

Adjacent sequences:  A235355 A235356 A235357 * A235359 A235360 A235361

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 07 2014

STATUS

approved

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Last modified September 17 13:00 EDT 2019. Contains 327131 sequences. (Running on oeis4.)