A366914 - OEIS

%I #24 Dec 27 2023 14:31:01

%S 2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,30,31,32,37,41,42,43,47,49,

%T 53,59,60,61,64,67,70,71,73,79,81,83,84,89,90,97,101,102,103,107,109,

%U 113,121,125,127,128,131,132,137,139,140,149,150,151,157,163,167

%N Numbers expressible as the sum of their distinct prime factors raised to a natural exponent.

%C Each prime factor must appear exactly once in the sum.

%H Robert Israel, <a href="/A366914/b366914.txt">Table of n, a(n) for n = 1..10000</a>

%e 30 is a term because its distinct prime factors are 2, 3 and 5, and 30 = 2^1 + 3^1 + 5^2 = 2^4 + 3^2 + 5^1.

%e 42 is a term because its distinct prime factors are 2, 3 and 7, and 42 = 2^3 + 3^3 + 7^1 = 2^5 + 3^1 + 7^1.

%e 60 is a term because its distinct prime factors are 2, 3 and 5, and 60 = 2^5 + 3^1 + 5^2.

%p filter:= proc(n) local P,S,p,i;

%p   P:= numtheory:-factorset(n);

%p   S:= mul(add(x^(p^i),i=1..floor(log[p](n))),p=P);

%p   coeff(S,x,n) > 0

%p end proc:

%p select(filter, [$1..1000]); # _Robert Israel_, Dec 27 2023

%o (PARI) isok(n)={my(f=factor(n)[,1], m=n-vecsum(f)); polcoef(prod(k=1, #f, my(c=f[k]); sum(j=1, logint(m+c, c), x^(c^j-c)) , 1 + O(x*x^m)), m)} \\ _Andrew Howroyd_, Oct 27 2023

%K nonn

%O 1,1

%A _Tanmaya Mohanty_, Oct 27 2023

%E Terms a(43) and beyond from _Andrew Howroyd_, Oct 27 2023