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A236511 a(n) = |{0 < k < n: p = 3*phi(k) + phi(n-k) - 1, p + 2, p + 6 and p + 8 are all prime}|, where phi(.) is Euler's totient function. 2
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 0, 2, 2, 2, 2, 2, 2, 0, 2, 0, 4, 4, 2, 1, 3, 4, 2, 2, 3, 0, 1, 3, 2, 3, 1, 4, 4, 3, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,13

COMMENTS

Conjecture: a(n) > 0 for all n > 1075.

We have verified this for n up to 50000.

The above conjecture implies the well-known conjecture that there are infinitely many prime quadruplets (p, p + 2, p + 6, p + 8).

LINKS

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

EXAMPLE

a(10) = 1 since 3*phi(3) + phi(7) - 1 = 6 + 6 - 1 = 11, 11 + 2 = 13, 11 + 6 = 17 and 11 + 8 = 19 are all prime.

a(57) = 1 since 3*phi(31) + phi(26) - 1 = 90 + 12 - 1 = 101, 101 + 2 = 103, 101 + 6 = 107 and 101 + 8 = 109 are all prime.

MATHEMATICA

p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[n+6]&&PrimeQ[n+8]

f[n_, k_]:=3*EulerPhi[k]+EulerPhi[n-k]-1

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

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

CROSSREFS

Cf. A000010, A000040, A007530, A236508.

Sequence in context: A281772 A082886 A287179 * A235924 A097304 A136745

Adjacent sequences:  A236508 A236509 A236510 * A236512 A236513 A236514

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 27 2014

STATUS

approved

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Last modified August 3 17:26 EDT 2020. Contains 336200 sequences. (Running on oeis4.)