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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 (list; graph; refs; listen; history; text; internal format)
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

Zhi-Wei 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[n-k]], 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

Zhi-Wei Sun, Nov 22 2013

STATUS

approved

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Last modified March 18 22:11 EDT 2019. Contains 321305 sequences. (Running on oeis4.)