

A232270


Number of ways to write n = x + y (0 < x <= y) with sigma(x) + phi(y) prime, where sigma(x) is the sum of all (positive) divisors of x and phi is Euler's totient function.


8



0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 2, 5, 3, 3, 2, 3, 3, 3, 2, 3, 4, 3, 1, 2, 3, 4, 2, 2, 3, 3, 3, 6, 3, 3, 4, 5, 5, 4, 2, 2, 4, 5, 3, 5, 3, 3, 2, 4, 2, 6, 5, 2, 6, 3, 6, 5, 2, 6, 5, 4, 4, 6, 3, 1, 5, 4, 6, 6, 2, 8, 6, 5, 4, 5, 3, 5, 4, 6, 2, 3, 8, 3, 5, 6, 4, 5, 4, 3, 5, 3, 4, 7, 6, 6
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OFFSET

1,5


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 29, 70, 105.
(ii) Any integer n > 164 can be written as x + y (x, y > 0) with sigma(x) + sigma(y) prime.


LINKS

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


EXAMPLE

a(2) = 1 since 2 = 1 + 1 with sigma(1) + phi(1) = 2 prime.
a(29) = 1 since 29 = 1 + 28 with sigma(1) + phi(28) = 13 prime.
a(70) = 1 since 70 = 9 + 61 with sigma(9) + phi(61) = 73 prime.
a(105) = 1 since 105 = 4 + 101 with sigma(4) + phi(101) = 107 prime.


MATHEMATICA

f[n_]:=f[n]=Sum[If[Mod[n, d]==0, d, 0], {d, 1, n}]
a[n_]:=a[n]=Sum[If[PrimeQ[f[k]+EulerPhi[nk]], 1, 0], {k, 1, n/2}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A000010, A000040, A000203.
Sequence in context: A054125 A177227 A174373 * A191517 A303370 A082528
Adjacent sequences: A232267 A232268 A232269 * A232271 A232272 A232273


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Nov 22 2013


STATUS

approved



