The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A008475 If n = Product (p_j^k_j) then a(n) = Sum (p_j^k_j) (a(1) = 0 by convention). 58
 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 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 T. D. Noe and 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 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 5 15:27 EST 2022. Contains 358588 sequences. (Running on oeis4.)