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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 (list; graph; refs; listen; history; text; internal format)
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

Zhi-Wei 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[n-k], 1, 0], {k, 1, n-1}]

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

Zhi-Wei Sun, Feb 05 2014

STATUS

approved

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Last modified February 16 21:37 EST 2020. Contains 331975 sequences. (Running on oeis4.)