A326696 Numbers m with at least one divisor d > 1 such that sigma(d) divides m.

6, 12, 18, 24, 28, 30, 36, 42, 48, 54, 56, 60, 66, 72, 78, 84, 90, 96, 102, 108, 112, 114, 117, 120, 126, 132, 138, 140, 144, 150, 156, 162, 168, 174, 180, 182, 186, 192, 196, 198, 204, 210, 216, 222, 224, 228, 234, 240, 246, 252, 258, 264, 270, 276, 280, 282

Offset: 1

Author

Jaroslav Krizek, Aug 30 2019

Keywords

• nonn

Comments

All integers m contain at least one divisor d (number 1) such that sigma(d) divides m.

See A309253 for the smallest numbers m with n divisors d such that sigma(d) divides m for n >= 1.

Supersequence of A097603 (multiples of perfect numbers).

From Bernard Schott, Sep 04 2019: (Start)

If m = 6 * k with k >= 1, then 2 divides m and sigma(2) = 3 also divides m; so, the positive multiples of 6 belong to this sequence.

This sequence is generated by the primitive terms. A primitive term m is necessarily of the form d * sigma(d) where 1 < d < m is a divisor of m. The first few primitives are: 6, 28, 117, 182, ...

Some subsequences of such primitives, not exhaustive list:

1) d is prime p and m = p * sigma(p) = p * (p+1) is oblong.

For p = 2, 13, 19, 37, ..., we get 6, 182, 380, 1406, ...

2) d = p^2 with p prime, and m = p^2 * (p^2 + p + 1).

For p = 2, 3, 5, 7, ..., we get m = 28, 117, 775, 2793, ...

3) d = 2^(q-1) and m = 2^(q-1) * (2^q -1), with q prime in A000043 and 2^q - 1 is a Mersenne prime in A000668, then m is a perfect number in A000039.

For q prime = 2, 3, 5, 7, 13, ..., we get m = 6, 28, 496, 8128, 33550336, ... (End)

Examples

			Divisors d of 12: 1, 2, 3, 4, 6, 12; corresponding sigma(d):1, 3, 4, 7, 12, 28; sigma(d) divides 12 for 4 divisors d > 1: 2, 3 and 6.
		

Links

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

Crossrefs

Cf. A000203, A173441, A309253, A323652, A326697, A326698.

Subsequences: A008588 \ {0}, A097603.

Programs

Magma

[m: m in [1..10^5] | #[d: d in Divisors(m) | IsIntegral(m / SumOfDivisors(d) ) and d gt 1] gt 0];

Maple

filter:= proc(n) local d;
  uses numtheory;
  ormap(t -> n mod sigma(t) = 0, divisors(n) minus {1})
end proc:
select(filter, [$2..1000]); # Robert Israel, Oct 07 2019

Mathematica

aQ[n_] := AnyTrue[Rest @ Divisors[n], Divisible[n, DivisorSigma[1, #]] &]; Select[Range[282], aQ] (* Amiram Eldar, Aug 31 2019 *)

PARI

isok(m) = fordiv(m, d, if ((d>1) && (!(m % sigma(d))), return(1))); \\ Michel Marcus, Sep 03 2019

Formula

A173441(a(n)) > 1; A326697(a(n)) > 1; A326697(a(n)) > 1.