

A237531


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


1



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

1,6


COMMENTS

Conjecture: a(n) > 0 for all n > 5.
Clearly, this implies the twin prime conjecture.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..5000
Z.W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014


EXAMPLE

a(12) = 1 since 12 = 3 + 9 with phi(3*9)  1 = 17 and phi(3*9) + 1 = 19 both prime.
a(19) = 1 since 19 = 1 + 18 with phi(1*18)  1 = 5 and phi(1*18) + 1 = 7 both prime.
a(86) = 1 since 86 = 8 + 78 with phi(8*78)  1 = 191 and phi(8*78) + 1 = 193 both prime.


MATHEMATICA

p[n_]:=PrimeQ[EulerPhi[n]1]&&PrimeQ[EulerPhi[n]+1]
a[n_]:=Sum[If[p[k(nk)], 1, 0], {k, 1, (n1)/2}]
Table[a[n], {n, 1, 80}]


CROSSREFS

Cf. A000010, A000040, A001359, A006512, A014574, A072281, A233547, A234200, A237127, A237130, A237168, A237523.
Sequence in context: A050677 A058013 A223934 * A238504 A031356 A304522
Adjacent sequences: A237528 A237529 A237530 * A237532 A237533 A237534


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 09 2014


STATUS

approved



