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A234808 a(n) = |{0 < k < n: p = k + phi(n-k) and 2*n - p are both prime}|, where phi(.) is Euler's totient function. 3
0, 1, 1, 2, 2, 3, 2, 0, 3, 1, 2, 5, 2, 1, 5, 1, 2, 7, 2, 1, 4, 1, 2, 1, 4, 1, 4, 2, 4, 11, 4, 2, 3, 1, 5, 2, 3, 2, 6, 1, 5, 15, 4, 2, 9, 1, 6, 2, 5, 4, 6, 4, 4, 3, 8, 3, 6, 4, 7, 21, 2, 4, 7, 1, 7, 4, 6, 4, 6, 4, 8, 22, 7, 3, 13, 1, 10, 5, 3, 5, 7, 4, 9, 5, 10, 5, 8, 7, 7, 6, 8, 5, 6, 3, 8, 6, 7, 4, 8, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Conjecture: a(n) > 0 except for n = 1, 8.

Clearly, this implies Goldbach's conjecture.

LINKS

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

EXAMPLE

a(3) = 1 since 2 + phi(1) = 3 and 2*3 - 3 = 3 are both prime.

a(20) = 1 since 11 + phi(9) = 17 and 2*20 - 17 = 23 are both prime.

a(22) = 1 since 1 + phi(21) = 13 and 2*22 - 13 = 31 are both prime.

a(24) = 1 since 9 + phi(15) = 17 and 2*24 - 17 = 31 are both prime.

a(76) = 1 since 67 + phi(9) = 73 and 2*76 - 73 = 79 are both prime.

MATHEMATICA

f[n_, k_]:=k+EulerPhi[n-k]

p[n_, k_]:=PrimeQ[f[n, k]]&&PrimeQ[2n-f[n, k]]

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

Table[a[n], {n, 1, 100}]

CROSSREFS

Cf. A000010, A000040, A002372, A002375, A233547, A233918, A234200, A234246, A234360, A234470, A234475, A234514, A234567, A234615, A234694

Sequence in context: A271226 A292372 A098008 * A248079 A127638 A127639

Adjacent sequences:  A234805 A234806 A234807 * A234809 A234810 A234811

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Dec 30 2013

STATUS

approved

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Last modified December 9 22:27 EST 2019. Contains 329880 sequences. (Running on oeis4.)