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A236472 a(n) = |{0 < k < n: p = prime(k) + phi(n-k), prime(p) + 2 and prime(p) + 6 are all prime}|, where phi(.) is Euler's totient function. 3
0, 1, 1, 0, 1, 2, 2, 2, 2, 1, 2, 1, 2, 1, 0, 1, 1, 2, 3, 0, 1, 1, 1, 2, 0, 1, 1, 0, 0, 2, 0, 2, 2, 1, 0, 0, 3, 1, 2, 0, 2, 2, 2, 1, 0, 0, 4, 1, 0, 0, 0, 0, 5, 0, 1, 1, 1, 2, 1, 1, 3, 0, 0, 2, 2, 0, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 3, 3, 3, 2, 0, 0, 2, 1, 1, 3, 0, 0, 2, 0, 3, 0, 0, 1, 1, 0, 2, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Conjecture: a(n) > 0 for every n = 330, 331, ....

We have verified this for n up to 80000.

The conjecture implies that there are infinitely many prime triples of the form {prime(p), prime(p) + 2, prime(p) + 6} with p prime. See A236464 for such primes p.

LINKS

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

EXAMPLE

a(10) = 1 since prime(2) + phi(8) = 3 + 4 = 7, prime(7) + 2 = 17 + 2 = 19 and prime(7) + 6 = 23 are all prime.

a(877) = 1 since prime(784) + phi(877-784) = 6007 + 60 = 6067, prime(6067) + 2 = 60101 + 2 = 60103 and prime(6067) + 6 = 60107 are all prime.

MATHEMATICA

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

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

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

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

CROSSREFS

Cf. A000010, A000040, A022004, A236456, A236460, A236464, A236467, A236470.

Sequence in context: A180174 A172363 A181877 * A175357 A232800 A248380

Adjacent sequences:  A236469 A236470 A236471 * A236473 A236474 A236475

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 26 2014

STATUS

approved

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Last modified December 15 17:03 EST 2019. Contains 330000 sequences. (Running on oeis4.)