

A237168


Number of ways to write 2*n  1 = 2*p + q with p, q, phi(p+1)  1 and phi(p+1) + 1 all prime, where phi(.) is Euler's totient function.


6



0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 2, 2, 2, 3, 1, 3, 3, 2, 2, 4, 1, 1, 3, 2, 2, 3, 1, 1, 3, 2, 2, 2, 1, 2, 3, 2, 2, 4, 1, 4, 5, 2, 1, 6, 3, 3, 2, 3, 2, 5, 1, 2, 5, 3, 3, 4, 3, 2, 6, 4, 4, 5, 2, 3, 7, 2, 2
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OFFSET

1,13


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 12.
(ii) Any even number greater than 4 can be written as p + q with p, q, phi(p+2)  1 and phi(p+2) + 1 all prime.
Part (i) implies both Lemoine's conjecture (cf. A046927) and the twin prime conjecture, while part (ii) unifies Goldbach's conjecture and the twin prime conjecture.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Unification of Goldbach's conjecture and the twin prime conjecture, a message to Number Theory List, Jan. 29, 2014.


EXAMPLE

a(9) = 1 since 2*9  1 = 2*7 + 3 with 7, 3, phi(7+1)  1 = 3 and phi(7+1) + 1 = 5 all prime.
a(934) = 1 since 2*934  1 = 2*457 + 953 with 457, 953, phi(457+1)  1 = 227 and phi(457+1) + 1 = 229 all prime.


MATHEMATICA

PQ[n_]:=PrimeQ[EulerPhi[n]1]&&PrimeQ[EulerPhi[n]+1]
a[n_]:=Sum[If[PQ[Prime[k]+1]&&PrimeQ[2n12*Prime[k]], 1, 0], {k, 1, PrimePi[n1]}]
Table[a[n], {n, 1, 70}]


CROSSREFS

Cf. A000010, A000040, A001359, A002375, A002372, A006512, A046927, A072281, A236566, A237127, A237130.
Sequence in context: A301304 A005086 A358333 * A350959 A157372 A270559
Adjacent sequences: A237165 A237166 A237167 * A237169 A237170 A237171


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 04 2014


STATUS

approved



