A362864 Numbers k that divide Sum_{i=1..k} (i - d(i)), where d(n) is the number of divisors of n (A000005).

1, 2, 5, 8, 15, 24, 26, 47, 121, 204, 347, 562, 4204, 6937, 6947, 31108, 379097, 379131, 379133, 2801205, 12554202, 20698345, 56264197, 13767391064, 37423648626, 37423648726, 61701166395, 276525443156, 276525443176, 455913379395, 455913379831, 751674084802

Offset: 1

Author

Ctibor O. Zizka, May 06 2023

Keywords

• nonn

Comments

Numbers k such that the mean number of nondivisors in the range 1..k is an integer.

Numbers k such that A161664(k) is divisible by k.

Numbers k such that (A000217(k) - A006218(k)) is divisible by k.

The subsequence of odd terms k equals the intersection of A050226 and this sequence.

Examples

			k = 5: Sum_{i=1..5} (i - d(i))/k = 5/5 = 1, so k = 5 is a term.
		

Links

• Martin Ehrenstein, Table of n, a(n) for n = 1..38

Crossrefs

Cf. A000005, A000217, A006218, A049820, A050226, A161664.

Programs

Mathematica

seq[kmax_] := Module[{sum = 0, s = {}}, Do[sum += k - DivisorSigma[0, k]; If[Divisible[sum, k], AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^6] (* Amiram Eldar, May 06 2023 *)

PARI

isok(k) = !(sum(i=1, k, i - numdiv(i)) % k); \\ Michel Marcus, May 06 2023

Python

from itertools import count, islice
from sympy import divisor_count
def A362864_gen(): # generator of terms
    c = 0
    for k in count(1):
        if not (c:=c+k-divisor_count(k))%k:
            yield k
A362864_list = list(islice(A362864_gen(),15)) # Chai Wah Wu, May 20 2023

Extensions

More terms from Amiram Eldar, May 06 2023

a(24)-a(32) from Martin Ehrenstein, May 22 2023