

A237184


Number of ordered ways to write n = (1+(n mod 2))*p + q with p, q, phi(p+1)  1 and phi(q1) + 1 all prime.


1



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

1,14


COMMENTS

Conjecture: a(n) > 0 for all n > 23.
This is stronger than Goldbach's conjecture and Lemoine's conjecture (cf. A046927).
We have verified the conjecture for n up to 3*10^6.


LINKS

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


EXAMPLE

a(10) = 1 since 10 = 7 + 3 with 7, 3, phi(7+1)  1 = 3 and phi(31) + 1 = 2 all prime.
a(499) = 1 since 499 = 2*199 + 101 with 199, 101, phi(199+1)  1 = 79 and phi(1011) + 1 = 41 all prime.
a(869) = 1 since 869 = 2*433 + 3 with 433, 3, phi(433+1)  1 = 179 and phi(31) + 1 = 2 all prime.


MATHEMATICA

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


CROSSREFS

Cf. A000040, A002372, A002375, A039698, A046927, A078892, A237127, A237130, A237168, A237183.
Sequence in context: A028932 A076473 A163160 * A029240 A302642 A025803
Adjacent sequences: A237181 A237182 A237183 * A237185 A237186 A237187


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 04 2014


STATUS

approved



