A134599 Sum of digital sums (base 3) of the prime factors of n.

0, 2, 1, 4, 3, 3, 3, 6, 2, 5, 3, 5, 3, 5, 4, 8, 5, 4, 3, 7, 4, 5, 5, 7, 6, 5, 3, 7, 3, 6, 3, 10, 4, 7, 6, 6, 3, 5, 4, 9, 5, 6, 5, 7, 5, 7, 5, 9, 6, 8, 6, 7, 7, 5, 6, 9, 4, 5, 5, 8, 5, 5, 5, 12, 6, 6, 5, 9, 6, 8, 7, 8, 5, 5, 7, 7, 6, 6, 7, 11, 4, 7, 3, 8, 8, 7, 4, 9, 5, 7, 6, 9, 4, 7, 6, 11, 5, 8, 5, 10, 5, 8, 5, 9, 7

Offset: 1

Author

Hieronymus Fischer, Nov 11 2007

Keywords

• nonn
• base
• easy

Examples

			a(6) = 3, since 6 = 2*3 and so a(6) = ds_3(2) + ds_3(3) = 2 + 1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A053735, A080773 (base 2), A118503.

Programs

Mathematica

f[p_, e_] := e * DigitSum[p, 3]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 30 2025 *)

PARI

a(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2] * sumdigits(f[i, 1], 3));} \\ Amiram Eldar, Jul 30 2025

Formula

If p_1*p_2*p_3* ... *p_m = n is the unique prime factorization of n, then a(n) = Sum_{k=1..m} ds_3(p_k), where ds_3 is the digital sum base 3.

Totally additive with a(p) = A053735(p). - Amiram Eldar, Jul 30 2025

Extensions

a(1) = 0 prepended by Amiram Eldar, Jul 30 2025