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A237184 Number of ordered ways to write n = (1+(n mod 2))*p + q with p, q, phi(p+1) - 1 and phi(q-1) + 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 (list; graph; refs; listen; history; text; internal format)
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

Zhi-Wei 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(3-1) + 1 = 2 all prime.

a(499) = 1 since 499 = 2*199 + 101 with 199, 101, phi(199+1) - 1 = 79 and phi(101-1) + 1 = 41 all prime.

a(869) = 1 since 869 = 2*433 + 3 with 433, 3, phi(433+1) - 1 = 179 and phi(3-1) + 1 = 2 all prime.

MATHEMATICA

pq[n_]:=PrimeQ[n]&&PrimeQ[EulerPhi[n+1]-1]

PQ[n_]:=PrimeQ[n]&&PrimeQ[EulerPhi[n-1]+1]

a[n_]:=Sum[If[pq[k]&&PQ[n-(1+Mod[n, 2])k], 1, 0], {k, 1, (n-1)/(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

Zhi-Wei Sun, Feb 04 2014

STATUS

approved

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Last modified February 17 17:27 EST 2019. Contains 320222 sequences. (Running on oeis4.)