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A236419 a(n) = |{0 < k < n: r = phi(k) + phi(n-k)/6 + 1 and p(r) + q(r) are both prime}|, where phi(.) is Euler's totient function, p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009). 4
0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 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, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 2, 1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 4, 1, 2, 1, 0, 2, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,68

COMMENTS

Conjecture: a(n) > 0 for all n > 127.

We have verified this for n up to 30000.

The conjecture implies that there are infinitely many primes r with p(r) + q(r) also 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 [math.NT], 2014.

EXAMPLE

a(15) = 1 since phi(1) + phi(14)/6 + 1 = 3 with p(3) + q(3) = 3 + 2 = 5 prime.

a(54) = 1 since phi(41) + phi(13)/6 + 1 = 43 with p(43) + q(43) = 63261 + 1610 = 64871 prime.

MATHEMATICA

pq[n_]:=PrimeQ[n]&&PrimeQ[PartitionsP[n]+PartitionsQ[n]]

f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/6+1

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

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

CROSSREFS

Cf. A000009, A000010, A000040, A232504, A236412, A236417.

Sequence in context: A302721 A098082 A321921 * A114448 A068933 A015472

Adjacent sequences:  A236416 A236417 A236418 * A236420 A236421 A236422

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 25 2014

STATUS

approved

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Last modified August 18 13:12 EDT 2019. Contains 326100 sequences. (Running on oeis4.)