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A237531 a(n) = |{0 < k < n/2: phi(k*(n-k)) - 1 and phi(k*(n-k)) + 1 are both prime}|, where phi(.) is Euler's totient function. 1
0, 0, 0, 0, 0, 2, 2, 2, 3, 2, 3, 1, 3, 1, 4, 1, 1, 1, 1, 4, 3, 2, 1, 2, 1, 1, 3, 2, 5, 4, 3, 3, 6, 2, 3, 1, 5, 4, 5, 2, 3, 5, 5, 3, 7, 6, 3, 7, 3, 8, 7, 4, 4, 5, 6, 4, 3, 9, 6, 8, 9, 8, 9, 9, 10, 7, 6, 3, 5, 4, 8, 4, 8, 5, 10, 2, 7, 9, 5, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Conjecture: a(n) > 0 for all n > 5.

Clearly, this implies the twin prime conjecture.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

EXAMPLE

a(12) = 1 since 12 = 3 + 9 with phi(3*9) - 1 = 17 and phi(3*9) + 1 = 19 both prime.

a(19) = 1 since 19 = 1 + 18 with phi(1*18) - 1 = 5 and phi(1*18) + 1 = 7 both prime.

a(86) = 1 since 86 = 8 + 78 with phi(8*78) - 1 = 191 and phi(8*78) + 1 = 193 both prime.

MATHEMATICA

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

a[n_]:=Sum[If[p[k(n-k)], 1, 0], {k, 1, (n-1)/2}]

Table[a[n], {n, 1, 80}]

CROSSREFS

Cf. A000010, A000040, A001359, A006512, A014574, A072281, A233547, A234200, A237127, A237130, A237168, A237523.

Sequence in context: A050677 A058013 A223934 * A238504 A031356 A304522

Adjacent sequences:  A237528 A237529 A237530 * A237532 A237533 A237534

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 09 2014

STATUS

approved

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Last modified June 20 10:13 EDT 2019. Contains 324234 sequences. (Running on oeis4.)