A326696 - OEIS

%I #28 Sep 08 2022 08:46:24

%S 6,12,18,24,28,30,36,42,48,54,56,60,66,72,78,84,90,96,102,108,112,114,

%T 117,120,126,132,138,140,144,150,156,162,168,174,180,182,186,192,196,

%U 198,204,210,216,222,224,228,234,240,246,252,258,264,270,276,280,282

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

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

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

%C Supersequence of A097603 (multiples of perfect numbers).

%C From _Bernard Schott_, Sep 04 2019: (Start)

%C 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.

%C 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, ...

%C Some subsequences of such primitives, not exhaustive list:

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

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

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

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

%C 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.

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

%H Robert Israel, <a href="/A326696/b326696.txt">Table of n, a(n) for n = 1..10000</a>

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

%e 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.

%p filter:= proc(n) local d;

%p   uses numtheory;

%p   ormap(t -> n mod sigma(t) = 0, divisors(n) minus {1})

%p end proc:

%p select(filter, [$2..1000]); # _Robert Israel_, Oct 07 2019

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

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

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

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

%Y Subsequences: A008588 \ {0}, A097603.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Aug 30 2019