

A237253


Number of ordered ways to write n = k + m with k > 0 and m > 0 such that phi(k)  1, phi(k) + 1 and prime(prime(prime(m)))  2 are all prime, where phi(.) is Euler's totient function.


3



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

1,8


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 7.
(ii) Any integer n > 22 can be written as k + m with k > 0 and m > 0 such that prime(k) + 2 and prime(prime(prime(m)))  2 are both prime.
Note that either part of the conjecture implies the twin prime conjecture.


LINKS

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


EXAMPLE

a(12) = 1 since 12 = 9 + 3 with phi(9)  1 = 5, phi(9) + 1 = 7 and prime(prime(prime(3)))  2 = prime(prime(5))  2 = prime(11)  2 = 29 all prime.
a(103) = 1 since 103 = 73 + 30 with phi(73)  1 = 71, phi(73) + 1 = 73 and prime(prime(prime(30)))  2 = prime(prime(113))  2 = prime(617)  2 = 4547 all prime.


MATHEMATICA

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


CROSSREFS

Cf. A000010, A000040, A001359, A006512, A072281, A218829, A236531, A236566, A237127, A237130, A237168.
Sequence in context: A049710 A025143 A174314 * A080634 A109925 A306260
Adjacent sequences: A237250 A237251 A237252 * A237254 A237255 A237256


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 05 2014


STATUS

approved



