A337310 Additive function with a(p) = p, a(p^e) = p*a(e) for prime p and e > 1, with a(1) = 1.
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, 10, 31, 10, 14, 19, 12, 10, 37, 21, 16, 11, 41, 12, 43, 15, 11, 25, 47, 11, 14, 12, 20, 17, 53, 11, 16, 13, 22, 31, 59, 12, 61, 33, 13, 10, 18, 16, 67, 21, 26, 14, 71, 12, 73
Offset: 1
Keywords
Examples
a(100) = a(2^2*5^2) = 2*a(2) + 5*a(2) = 2*2 + 5*2 = 14. 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.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..65537
Programs
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Maple
a:= proc(n) option remember; `if`(n=1, 1, add(i[1]*a(i[2]), i=ifactors(n)[2])) end: seq(a(n), n=1..100); # Alois P. Heinz, Aug 22 2020 -
Mathematica
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 *)
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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
Formula
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.
Comments