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A008475 If n = Product (p_j^k_j) then a(n) = Sum (p_j^k_j) (a(1) = 0 by convention). 51
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 (list; graph; refs; listen; history; text; internal format)
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

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

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.

LINKS

T. D. Noe and Daniel Forgues, Table of n, a(n) for n=1..100000 (first 10000 terms from T. D. Noe)

J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 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(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

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, A081402-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 A082081 * A222416 A269524 A161656 A225090

Adjacent sequences:  A008472 A008473 A008474 * A008476 A008477 A008478

KEYWORD

nonn,nice

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified December 11 13:46 EST 2017. Contains 295876 sequences.