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a^n + b^n + c^n + ..., where a*b*c* ... is the prime factorization of n.
4

%I #26 Dec 15 2017 17:36:11

%S 1,4,27,32,3125,793,823543,768,39366,9766649,285311670611,539633,

%T 302875106592253,678223089233,30531927032,262144,

%U 827240261886336764177,775103122,1978419655660313589123979,95367433737777,558545874543637210

%N a^n + b^n + c^n + ..., where a*b*c* ... is the prime factorization of n.

%C n*log_10(2) + log_10(log_2(n)) <= length(a(n)) <= n*log_10(n). - Martin Renner, Jan 18 2012

%C If m = p^k is a power of a prime then a(n) = sum(p^m,i=1..k) = k*p^m is composite. - Martin Renner, Jan 31 2013

%H T. D. Noe, <a href="/A082872/b082872.txt">Table of n, a(n) for n = 1..100</a>

%e a(6) = a(2*3) = 2^6 + 3^6 = 793.

%e a(8) = a(2*2*2) = 2^8 + 2^8 + 2^8 = 768.

%p A082872 := proc(n)

%p local ps;

%p if n= 1 then

%p 1;

%p else

%p ps := ifactors(n)[2] ;

%p add( op(2,p)*op(1,p)^n,p=ps) ;

%p end if;

%p end proc: # _R. J. Mathar_, Mar 12 2014

%t Table[f = FactorInteger[n]; Total[Flatten[Table[Table[f[[i, 1]], {f[[i, 2]]}], {i, Length[f]}]]^n], {n, 25}] (* _T. D. Noe_, Feb 01 2013 *)

%t Table[Total[Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[n]]^n],{n,30}] (* _Harvey P. Dale_, Jun 10 2016 *)

%Y Cf. A082813, A082814, A051674.

%K nonn,easy

%O 1,2

%A _Jason Earls_, May 25 2003