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A337310 Additive function with a(p) = p, a(p^e) = p*a(e) for prime p and e > 1, with a(1) = 1.

%I A337310 #52 Jul 21 2025 11:19:31
%S A337310 1,2,3,4,5,5,7,6,6,7,11,7,13,9,8,8,17,8,19,9,10,13,23,9,10,15,9,11,29,
%T A337310 10,31,10,14,19,12,10,37,21,16,11,41,12,43,15,11,25,47,11,14,12,20,17,
%U A337310 53,11,16,13,22,31,59,12,61,33,13,10,18,16,67,21,26,14,71,12,73
%N A337310 Additive function with a(p) = p, a(p^e) = p*a(e) for prime p and e > 1, with a(1) = 1.
%C A337310 a(n) <= A001414(n) for n > 1, with equality if and only if all the exponents in the prime factorization of n are either less than 6 or prime themselves. - _Mital Ashok_, Jun 22 2025
%H A337310 Antti Karttunen, <a href="/A337310/b337310.txt">Table of n, a(n) for n = 1..65537</a>
%F A337310 a(1)=1, a(p_1^b_1*p_2^b_2*...*p_n^b_n)=p_1*a(b_1)+p_2*a(b_2)+...+p_n*a(b_n) where p_i is the i-th prime number.
%e A337310 a(100) = a(2^2*5^2) = 2*a(2) + 5*a(2) = 2*2 + 5*2 = 14.
%e A337310 a(192) = a(2^6*3^1) = 2*a(6) + 3*a(1) = 2*a(2^1*3^1) + 3*1 = 2*(2*a(1) + 3*a(1)) + 3 = 2*(2*1 + 3*1) + 3 = 13.
%p A337310 a:= proc(n) option remember; `if`(n=1, 1,
%p A337310       add(i[1]*a(i[2]), i=ifactors(n)[2]))
%p A337310     end:
%p A337310 seq(a(n), n=1..100);  # _Alois P. Heinz_, Aug 22 2020
%t A337310 f[p_, e_] := p * a[e]; a[1] = 1; a[n_] := a[n] = Plus @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Aug 22 2020 *)
%o A337310 (PARI) a(n)={my(f=factor(n)); if(n==1, 1, sum(i=1, #f~, my([p,e]=f[i,]); p*a(e)))} \\ _Andrew Howroyd_, Aug 22 2020
%Y A337310 Cf. A000040, A001414, A064372, A279513.
%K A337310 nonn
%O A337310 1,2
%A A337310 Ferdinand Rönngren and _Lars Kevin Haagensen Strömberg_, Aug 22 2020