A008475 If n = Product (p_j^k_j) then a(n) = Sum (p_j^k_j) (a(1) = 0 by convention).

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, 11, 29, 10, 31, 32, 14, 19, 12, 13, 37, 21, 16, 13, 41, 12, 43, 15, 14, 25, 47, 19, 49, 27, 20, 17, 53, 29, 16, 15, 22, 31, 59, 12, 61, 33, 16, 64, 18, 16, 67, 21, 26, 14, 71, 17, 73

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.

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

a(n) = Sum_{k=1..A001221(n)} A027748(n,k) ^ A124010(n,k) for n>1. - Reinhard Zumkeller, Oct 10 2011

a(n) = Sum_{k=1..A001221(n)} A141809(n,k) for n > 1. - Reinhard Zumkeller, Jan 29 2013

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

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

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

Sequence in context: A156229 A340901 A082081 * A222416 A269524 A161656

Adjacent sequences: A008472 A008473 A008474 * A008476 A008477 A008478

Keyword

nonn,nice

Author

Olivier Gérard

Status

approved