OFFSET
1,2
COMMENTS
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
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
REFERENCES
F. J. Budden, The Fascination of Groups, Cambridge, 1972; pp. 322, 573.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter IV, p. 147.
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.
LINKS
Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
John Bamberg, Grant Cairns and Devin Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, Vol. 110, No. 3 (March 2003), pp. 202-209.
Roger B. Eggleton and William P. Galvin, Upper Bounds on the Sum of Principal Divisors of an Integer, Mathematics Magazine, Vol. 77, No. 3 (Jun., 2004), pp. 190-200.
FORMULA
Additive with a(p^e) = p^e.
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
EXAMPLE
a(180) = a(2^2 * 3^2 * 5) = 2^2 + 3^2 + 5 = 18.
MAPLE
A008475 := proc(n) local e, j; e := ifactors(n)[2]:
add(e[j][1]^e[j][2], j=1..nops(e)) end:
seq(A008475(n), n=1..60); # Peter Luschny, Jan 17 2010
MATHEMATICA
f[n_] := Plus @@ Power @@@ FactorInteger@ n; f[1] = 0; Array[f, 73]
PROG
(PARI) for(n=1, 100, print1(sum(i=1, omega(n), component(component(factor(n), 1), i)^component(component(factor(n), 2), i)), ", "))
(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 */
(PARI) A008475(n) = { my(f=factor(n)); vecsum(vector(#f~, i, f[i, 1]^f[i, 2])); }; \\ Antti Karttunen, Nov 17 2017
(Haskell)
a008475 1 = 0
a008475 n = sum $ a141809_row n
-- Reinhard Zumkeller, Jan 29 2013, Oct 10 2011
(Python)
from sympy import factorint
def a(n):
f=factorint(n)
return 0 if n==1 else sum([i**f[i] for i in f]) # Indranil Ghosh, May 20 2017
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
STATUS
approved