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A326696 Numbers m with at least one divisor d > 1 such that sigma(d) divides m. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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)

LINKS

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

FORMULA

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

EXAMPLE

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.

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 *)

PROG

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

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

CROSSREFS

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

Subsequences: A008588 \ {0}, A097603.

Sequence in context: A037917 A282146 A204879 * A097603 A315754 A315755

Adjacent sequences:  A326693 A326694 A326695 * A326697 A326698 A326699

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Aug 30 2019

STATUS

approved

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Last modified January 19 14:53 EST 2020. Contains 331049 sequences. (Running on oeis4.)