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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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: A050949 A074943 A272011 * A045719 A114906 A259179

Adjacent sequences:  A236371 A236372 A236373 * A236375 A236376 A236377

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 24 2014

STATUS

approved

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Last modified August 8 02:22 EDT 2020. Contains 336290 sequences. (Running on oeis4.)