%I #76 Oct 30 2023 07:15:03

%S 0,2,3,4,5,5,7,8,9,7,11,7,13,9,8,16,17,11,19,9,10,13,23,11,25,15,27,

%T 11,29,10,31,32,14,19,12,13,37,21,16,13,41,12,43,15,14,25,47,19,49,27,

%U 20,17,53,29,16,15,22,31,59,12,61,33,16,64,18,16,67,21,26,14,71,17,73

%N If n = Product (p_j^k_j) then a(n) = Sum (p_j^k_j) (a(1) = 0 by convention).

%C For n>1, a(n) is the minimal number m such that the symmetric group S_m has an element of order n. - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 26 2001

%C If gcd(u,w) = 1, then a(u*w) = a(u) + a(w); behaves like logarithm; compare A001414 or A056239. - _Labos Elemer_, Mar 31 2003

%D F. J. Budden, The Fascination of Groups, Cambridge, 1972; pp. 322, 573.

%D József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter IV, p. 147.

%D T. Z. Xuan, On some sums of large additive number theoretic functions (in Chinese), Journal of Beijing normal university, No. 2 (1984), pp. 11-18.

%H Daniel Forgues, <a href="/A008475/b008475.txt">Table of n, a(n) for n = 1..100000</a> (first 10000 terms from T. D. Noe)

%H John Bamberg, Grant Cairns and Devin Kilminster, <a href="http://www.jstor.org/stable/3647934">The crystallographic restriction, permutations and Goldbach's conjecture</a>, Amer. Math. Monthly, Vol. 110, No. 3 (March 2003), pp. 202-209.

%H Roger B. Eggleton and William P. Galvin, <a href="http://www.jstor.org/stable/3219115">Upper Bounds on the Sum of Principal Divisors of an Integer</a>, Mathematics Magazine, Vol. 77, No. 3 (Jun., 2004), pp. 190-200.

%F Additive with a(p^e) = p^e.

%F a(A000961(n)) = A000961(n); a(A005117(n)) = A001414(A005117(n)).

%F a(n) = Sum_{k=1..A001221(n)} A027748(n,k) ^ A124010(n,k) for n>1. - _Reinhard Zumkeller_, Oct 10 2011

%F a(n) = Sum_{k=1..A001221(n)} A141809(n,k) for n > 1. - _Reinhard Zumkeller_, Jan 29 2013

%F Sum_{k=1..n} a(k) ~ (Pi^2/12)* n^2/log(n) + O(n^2/log(n)^2) (Xuan, 1984). - _Amiram Eldar_, Mar 04 2021

%e a(180) = a(2^2 * 3^2 * 5) = 2^2 + 3^2 + 5 = 18.

%p A008475 := proc(n) local e,j; e := ifactors(n)[2]:

%p add(e[j][1]^e[j][2], j=1..nops(e)) end:

%p seq(A008475(n), n=1..60); # _Peter Luschny_, Jan 17 2010

%t f[n_] := Plus @@ Power @@@ FactorInteger@ n; f[1] = 0; Array[f, 73]

%o (PARI) for(n=1,100,print1(sum(i=1,omega(n), component(component(factor(n),1),i)^component(component(factor(n),2),i)),","))

%o (PARI) a(n)=local(t);if(n<1,0,t=factor(n);sum(k=1,matsize(t)[1],t[k,1]^t[k,2])) /* _Michael Somos_, Oct 20 2004 */

%o (PARI) A008475(n) = { my(f=factor(n)); vecsum(vector(#f~,i,f[i,1]^f[i,2])); }; \\ _Antti Karttunen_, Nov 17 2017

%o (Haskell)

%o a008475 1 = 0

%o a008475 n = sum $ a141809_row n

%o -- _Reinhard Zumkeller_, Jan 29 2013, Oct 10 2011

%o (Python)

%o from sympy import factorint

%o def a(n):

%o f=factorint(n)

%o return 0 if n==1 else sum([i**f[i] for i in f]) # _Indranil Ghosh_, May 20 2017

%Y Cf. A001414, A000961, A005117, A051613, A072691, A081402, A081403, A081404, A027748, A124010, A001221, A028233, A034684, A053585, A159077, A023888, A078771, A092509, A286875.

%Y See A222416 for the variant with a(1)=1.

%K nonn,nice

%O 1,2

%A _Olivier Gérard_