A345068 a(n) = Sum_{d|n, d>1} d^floor(1/omega(d)).

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, 39, 15, 29, 14, 31, 62, 15, 20, 13, 22, 37, 22, 17, 22, 41, 16, 43, 19, 19, 26, 47, 37, 56, 34, 21, 21, 53, 44, 17, 24, 23, 32, 59, 21, 61, 34, 21, 126, 19, 20, 67, 25, 27, 18, 71, 32

Offset

1,2

Comments

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.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

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

If p is prime, a(p^m) = (p^(m+1)-p)/(p-1). - Robert Israel, Oct 09 2024

Example

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

Maple

f:= proc(n) local t, D1, D2;
  D1, D2:= selectremove(t -> nops(numtheory:-factorset(t))<= 1, numtheory:-divisors(n) minus {1});
  convert(D1, `+`) + nops(D2)
end proc:
map(f, [$1..100]); # Robert Israel, Oct 09 2024

Mathematica

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

Prog

(PARI) a(n) = sumdiv(n, d, if (d>1, if (omega(d)==1, d, 1))); \\ Michel Marcus, Oct 09 2024

Crossrefs

Cf. A000961, A001221 (omega).

Sequence in context: A369319 A336465 A340774 * A057723 A335835 A361480

Adjacent sequences: A345065 A345066 A345067 * A345069 A345070 A345071

Keyword

nonn,look

Author

Wesley Ivan Hurt, Jun 06 2021

Status

approved