

A236374


a(n) = {0 < k < n: m = phi(k)/2 + phi(nk)/8 is an integer with 2^(m1)*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
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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^(m1)*phi(m)  1 prime.
See A236375 for a list of known numbers m with 2^(m1)*phi(m)  1 prime.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000


EXAMPLE

a(24) = 1 since phi(8)/2 + phi(16)/8 = 3 with 2^(31)*phi(3)  1 = 7 prime.
a(33) = 1 since phi(13)/2 + phi(20)/8 = 7 with 2^(71)*phi(7)  1 = 383 prime.
a(79) = 1 since phi(27)/2 + phi(52)/8 = 9 + 3 = 12 with 2^(121)*phi(12)  1 = 2^(13)  1 = 8191 prime.


MATHEMATICA

q[n_]:=IntegerQ[n]&&PrimeQ[2^(n1)*EulerPhi[n]1]
f[n_, k_]:=EulerPhi[k]/2+EulerPhi[nk]/8
a[n_]:=Sum[If[q[f[n, k]], 1, 0], {k, 1, n1}]
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

ZhiWei Sun, Jan 24 2014


STATUS

approved



