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

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei 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[2n-1-2*Prime[k]], 1, 0], {k, 1, PrimePi[n-1]}]

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

Zhi-Wei Sun, Feb 04 2014

STATUS

approved

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Last modified February 3 13:54 EST 2023. Contains 360035 sequences. (Running on oeis4.)