A345068 - OEIS

%I #13 Oct 09 2024 21:26:41

%S 0,2,3,6,5,6,7,14,12,8,11,11,13,10,9,30,17,16,19,13,11,14,23,20,30,16,

%T 39,15,29,14,31,62,15,20,13,22,37,22,17,22,41,16,43,19,19,26,47,37,56,

%U 34,21,21,53,44,17,24,23,32,59,21,61,34,21,126,19,20,67,25,27,18,71,32

%N a(n) = Sum_{d|n, d>1} d^floor(1/omega(d)).

%C For each nontrivial divisor of n, take a running total: add d if d is a prime power (i.e., if d = p^k where p is prime and k is a positive integer), otherwise add 1. For example, a(12) = 2 + 3 + 4 + 1 + 1 = 11.

%H Robert Israel, <a href="/A345068/b345068.txt">Table of n, a(n) for n = 1..10000</a>

%F If p is prime, a(p) = Sum_{d|p, d>1} d^floor(1/omega(d)) = p^floor(1/omega(p)) = p^1 = p.

%F If p is prime, a(p^m) = (p^(m+1)-p)/(p-1). - _Robert Israel_, Oct 09 2024

%e a(18) = Sum_{d|18, d>1} d^floor(1/omega(d)) = 2^1 + 3^1 + 6^0 + 9^1 + 18^0 = 16.

%p f:= proc(n) local t,D1,D2;

%p   D1,D2:= selectremove(t -> nops(numtheory:-factorset(t))<= 1, numtheory:-divisors(n) minus {1});

%p   convert(D1,`+`) + nops(D2)

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Oct 09 2024

%t Table[Sum[k^Floor[1/PrimeNu[k]] (1 - Ceiling[n/k] + Floor[n/k]), {k, 2, n}], {n, 100}]

%o (PARI) a(n) = sumdiv(n, d, if (d>1, if (omega(d)==1, d, 1))); \\ _Michel Marcus_, Oct 09 2024

%Y Cf. A000961, A001221 (omega).

%K nonn,look

%O 1,2

%A _Wesley Ivan Hurt_, Jun 06 2021