A168065 If n = Product p(k)^e(k) then a(n) = (Product (p(k)+1)^e(k) + Product (p(k)-1)^e(k))/2, a(1) = 1.

1, 2, 3, 5, 5, 7, 7, 14, 10, 11, 11, 19, 13, 15, 16, 41, 17, 26, 19, 29, 22, 23, 23, 55, 26, 27, 36, 39, 29, 40, 31, 122, 34, 35, 36, 74, 37, 39, 40, 83, 41, 54, 43, 59, 56, 47, 47, 163, 50, 62, 52, 69, 53, 100, 56, 111, 58, 59, 59, 112, 61, 63, 76, 365, 66, 82, 67, 89, 70, 84, 71

Offset: 1

Author

Daniel Forgues, Nov 18 2009

Keywords

• nonn

Comments

a(n) = n iff n is 1 or a prime;

a(n) = n+1 iff n is a biprime, i.e., n = p*q, p <= q primes;

a(n) = n+(p+q+r) iff n is a triprime, i.e., n = p*q*r, p <= q <= r primes;

a(n) = n + (p*q + p*r + p*s + q*r + q*s + r*s) + 1 iff n is a quadprime, i.e., n = p*q*r*s, p <= q <= r <= s primes;

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Links

• Daniel Forgues, Table of n, a(n) for n = 1..100000

Crossrefs

Cf. A003958, A003959, A168066.

Programs

PARI

a(n) = {f = factor(n); return ((prod(k=1, #f~, (f[k, 1]+1)^f[k, 2]) + prod(k=1, #f~, (f[k, 1]-1)^f[k, 2]))/2);} \\ Michel Marcus, Jun 13 2013

Formula

a(n) = (A003959(n) + A003958(n))/2.