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

%I A134599 #19 Jul 30 2025 09:46:54
%S A134599 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,
%T A134599 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,
%U A134599 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
%N A134599 Sum of digital sums (base 3) of the prime factors of n.
%H A134599 Amiram Eldar, <a href="/A134599/b134599.txt">Table of n, a(n) for n = 1..10000</a>
%F A134599 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.
%F A134599 Totally additive with a(p) = A053735(p). - _Amiram Eldar_, Jul 30 2025
%e A134599 a(6) = 3, since 6 = 2*3 and so a(6) = ds_3(2) + ds_3(3) = 2 + 1.
%t A134599 f[p_, e_] := e * DigitSum[p, 3]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jul 30 2025 *)
%o A134599 (PARI) a(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2] * sumdigits(f[i, 1], 3));} \\ _Amiram Eldar_, Jul 30 2025
%Y A134599 Cf. A053735, A080773 (base 2), A118503.
%K A134599 nonn,base,easy
%O A134599 1,2
%A A134599 _Hieronymus Fischer_, Nov 11 2007
%E A134599 a(1) = 0 prepended by _Amiram Eldar_, Jul 30 2025