

A237127


Number of ways to write n = k + m (0 < k < m) with k and m terms of A072281.


8



0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 3, 2, 3, 2, 3, 3, 3, 2, 2, 4, 3, 3, 2, 2, 4, 3, 4, 3, 4, 4, 3, 3, 4, 5, 4, 1, 3, 3, 5, 4, 4, 4, 4, 5, 3, 4, 2, 4, 4, 4, 5, 2, 4, 1, 4, 4, 4, 4, 1, 3, 4, 4, 5, 5
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OFFSET

1,15


COMMENTS

Conjecture: a(n) > 0 for all n > 11.
Clearly, this implies the twin prime conjecture.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000


EXAMPLE

a(13) = 1 since 13 = 5 + 8 with phi(5)  1 = 3, phi(5) + 1 = 5, phi(8)  1 = 3 and phi(8) + 1 = 5 all prime.
a(60) = 1 since 60 = 18 + 42 with phi(18)  1 = 5, phi(18) + 1 = 7, phi(42)  1 = 11 and phi(42) + 1 = 13 all prime.
a(84) = 1 since 84 = 7 + 77 with phi(7)  1 = 5, phi(7) + 1 = 7, phi(77)  1 = 59 and phi(77) + 1 = 61 all prime.


MATHEMATICA

PQ[n_]:=PrimeQ[EulerPhi[n]1]&&PrimeQ[EulerPhi[n]+1]
a[n_]:=Sum[If[PQ[k]&&PQ[nk], 1, 0], {k, 1, (n1)/2}]
Table[a[n], {n, 1, 70}]


CROSSREFS

Cf. A000010, A000040, A001359, A006512, A014574, A072281.
Sequence in context: A305030 A110917 A070956 * A262746 A007828 A070804
Adjacent sequences: A237124 A237125 A237126 * A237128 A237129 A237130


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 04 2014


STATUS

approved



