

A236097


a(n) = {0 < k < n2: p = phi(k) + phi(nk)/2 + 1, prime(p)  p  1 and prime(p)  p + 1 are all prime}, where phi(.) is Euler's totient function.


7



0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 3, 1, 1, 2, 2, 3, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 2, 0, 5, 5, 2, 4, 1, 5, 3, 3, 2, 4, 4, 9, 5, 9, 4, 10, 3, 6, 6, 8, 5, 10, 4, 4, 7, 8, 10, 5, 8, 9, 9, 4, 11, 3, 5, 5, 9, 5, 4, 4, 5, 6, 8, 7, 6, 3, 11, 4, 8, 10, 9, 8, 7, 6, 11, 7, 9, 4, 6, 5, 6, 2, 9, 4, 7, 6, 7, 10, 9
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OFFSET

1,8


COMMENTS

Conjecture: a(n) > 0 for all n > 31.
This implies that there are infinitely many primes p with {prime(p)  p  1, prime(p)  p + 1} a twin prime pair.


LINKS

ZhiWei 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(20) = 1 since phi(2) + phi(18)/2 + 1 = 5, prime(5)  5  1 = 5 and prime(5)  5 + 1 = 7 are all prime.
a(36) = 1 since phi(21) + phi(15)/2 + 1 = 17, prime(17)  17  1 = 41 and prime(17)  17 + 1 = 43 are all prime.


MATHEMATICA

p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]n1]&&PrimeQ[Prime[n]n+1]
f[n_, k_]:=EulerPhi[k]+EulerPhi[nk]/2+1
a[n_]:=Sum[If[p[f[n, k]], 1, 0], {k, 1, n3}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A000010, A000040, A001359, A006512, A014574, A234694, A234695, A235924, A236074, A236119.
Sequence in context: A264033 A236293 A056044 * A239319 A236468 A116685
Adjacent sequences: A236094 A236095 A236096 * A236098 A236099 A236100


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jan 19 2014


STATUS

approved



