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%I #7 Mar 21 2014 05:59:32
%S 4,27,256,1728,2401,3125,11664,72000,78732,200000,486000,531441,
%T 823543,1350000,3280500,9112500,22143375,52706752,56250000,61509375,
%U 156250000
%N Numbers n such that Sum_{i=1..j} 1/pn(i) + Sum_{i=1..k} 1/pd(i) is integer, where pn are the prime factors of n and pd the prime factors of the arithmetic derivative of n, both counted with multiplicity.
%e Arithmetic derivative of 2401 is 1372. Prime factors of 2401 are 7^4; prime factors of 1372 are 2^2, 7^3 and 1/7 + 1/7 + 1/7 + 1/7 + 1/2 + 1/2 + 1/7 + 1/7 + 1/7 = 2.
%p with(numtheory); P:= proc(q) local a,b,c,n,p;
%p for n from 2 to q do a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
%p b:=ifactors(a)[2]; c:=ifactors(n)[2]; if type(add(c[k][2]/c[k][1],k=1..nops(c))+add(b[k][2]/b[k][1],k=1..nops(b)),integer) then print(n); fi; od; end: P(10^9);
%Y Cf. A003415, A239491.
%K nonn,more
%O 1,1
%A _Paolo P. Lava_, Mar 20 2014
%E a(15)-a(21) from _Giovanni Resta_, Mar 20 2014