A236265 -id:A236265 - cp's OEIS frontend

A236263 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with m! + prime(m) prime}|, where phi(.) is Euler's totient function.

0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 0, 1, 2, 3, 3, 4, 5, 4, 4, 5, 7, 4, 5, 6, 6, 5, 5, 5, 7, 6, 7, 9, 7, 8, 7, 7, 5, 11, 8, 8, 8, 11, 8, 7, 5, 10, 6, 9, 8, 10, 7, 8, 10, 9, 7, 8, 9, 13, 8, 8, 9, 10, 6, 11, 10, 7, 7, 9, 11, 13, 8, 11, 13, 11, 14, 6

Offset: 1

Zhi-Wei Sun, Jan 21 2014

nonn

It seems that a(n) > 0 for all n > 17. (We have verified this for n up to 13000.) If a(n) > 0 infinitely often, then there are infinitely many positive integers m with m! + prime(m) prime.

See also A236265 for a similar sequence.

			a(18) = 1 since phi(3)/2 + phi(15)/8 = 1 + 1 = 2 with 2! + prime(2) = 2 + 3 = 5 prime.
a(356) = 1 since phi(203)/2 + phi(153)/8 = 84 + 12 = 96 with 96! + prime(96) = 96! + 503 prime.
a(457) = 1 since phi(7)/2 + phi(450)/8 = 3 + 15 = 18 with 18! + prime(18) = 18! + 61 = 6402373705728061 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..8100

Cf. A000010, A000040, A000142, A063499, A064278, A064401, A236241, A236256, A236265.

q[n_]:=IntegerQ[n]&&PrimeQ[n!+Prime[n]]
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}]

A236325 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/12 is an integer with m! + prime(m) prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 2, 1, 2, 3, 4, 3, 4, 4, 5, 2, 4, 3, 4, 5, 5, 6, 5, 6, 8, 7, 9, 8, 6, 6, 5, 8, 9, 4, 8, 7, 7, 5, 5, 7, 7, 8, 8, 6, 7, 8, 7, 10, 5, 8, 9, 8, 7, 7, 6, 7, 8, 12, 10, 6, 8, 9, 9, 12, 9, 8, 7, 13

Offset: 1

Zhi-Wei Sun, Jan 22 2014

nonn

It might seem that a(n) > 0 for all n > 14, but a(7365) = 0. If a(n) > 0 infinitely often, then there are infinitely many positive integers m with m! + prime(m) prime.

			a(10) = 1 since phi(1)/2 + phi(9)/12 = 1/2 + 6/12 = 1 with 1! + prime(1) = 1 + 2 = 3 prime.
a(23) = 1 since phi(10)/2 + phi(13)/12 = 2 + 1 = 3 with 3! + prime(3) = 6 + 5 = 11 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..7000

Cf. A000010, A000040, A000142, A063499, A064278, A064401, A236241, A236256, A236263, A236265.

p[n_]:=IntegerQ[n]&&PrimeQ[n!+Prime[n]]
f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/12
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A236263 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with m! + prime(m) prime}|, where phi(.) is Euler's totient function.

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