A236374 - OEIS

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A236374

a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with 2^(m-1)*phi(m) - 1 prime}|, where phi(.) is Euler's totient function.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 1, 2, 2, 2, 4, 2, 1, 0, 2, 1, 2, 3, 3, 3, 4, 2, 2, 2, 3, 5, 3, 4, 4, 1, 1, 2, 3, 7, 4, 3, 5, 3, 3, 2, 4, 5, 4, 3, 4, 3, 2, 6, 7, 5, 5, 4, 4, 5, 4, 5, 5, 3, 7, 3, 5, 1, 7, 4, 7, 7, 5, 9, 5, 9, 3, 3, 5, 13, 7, 9, 7, 3, 4, 10, 10, 9, 11

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OFFSET

1,20

COMMENTS

Conjecture: a(n) > 0 for all n > 31.

We have verified this for n up to 60000.

The conjecture implies that there are infinitely many positive integers m with 2^(m-1)*phi(m) - 1 prime.

See A236375 for a list of known numbers m with 2^(m-1)*phi(m) - 1 prime.

LINKS

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

EXAMPLE

a(24) = 1 since phi(8)/2 + phi(16)/8 = 3 with 2^(3-1)*phi(3) - 1 = 7 prime.

a(33) = 1 since phi(13)/2 + phi(20)/8 = 7 with 2^(7-1)*phi(7) - 1 = 383 prime.

a(79) = 1 since phi(27)/2 + phi(52)/8 = 9 + 3 = 12 with 2^(12-1)*phi(12) - 1 = 2^(13) - 1 = 8191 prime.

MATHEMATICA

q[n_]:=IntegerQ[n]&&PrimeQ[2^(n-1)*EulerPhi[n]-1]

f[n_, k_]:=EulerPhi[k]/2+EulerPhi[n-k]/8

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

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

CROSSREFS

Cf. A000010, A000040, A000079, A236375.

Sequence in context: A074943 A272011 A352362 * A045719 A114906 A259179

Adjacent sequences: A236371 A236372 A236373 * A236375 A236376 A236377

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 24 2014

STATUS

approved