A326697 -id:A326697 - cp's OEIS frontend

A326698 a(n) is the product of divisors d of n such that sigma(d) divides n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 36, 1, 1, 1, 4, 1, 10, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 2, 1, 784, 1, 1, 1, 180, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Jaroslav Krizek, Jul 19 2019

nonn

a(A097603(n)) > 1.

See A173441 and A326697 for number and sum such divisors.

			For n = 12, divisors d of 12: 1, 2, 3, 4, 6, 12;
corresponding sigma(d): 1, 3, 4, 7, 12, 28;
sigma(d) divides n for 4 divisors d: 1, 2, 3, 6;
a(12) = 1 * 2 * 3 * 6 = 36.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A000961, A173441, A326697.

[&*[d: d in Divisors(n) | IsIntegral(n / SumOfDivisors(d))]: n in [1..100]];

a[n_] := Times @@ Select[Divisors[n], Divisible[n, DivisorSigma[1, #] &]]; Array[a, 100] (* Amiram Eldar, Jul 21 2019 *)

a(n) = my(p=1); fordiv(n, d, if (!(n % sigma(d)), p *= d)); p; \\ Michel Marcus, Jul 19 2019

A309253 a(n) is the smallest number m with exactly n such divisors d that sigma(d) divides m.

Original entry on oeis.org

1, 6, 30, 12, 60, 84, 1140, 120, 168, 2340, 1848, 360, 2184, 1080, 4368, 840, 10440, 1680, 7920, 2520, 6552, 3360, 7560, 5040, 13104, 27720, 73440, 36960, 21840, 15120, 72072, 10080, 95760, 26208, 63840, 20160, 146160, 144144, 87360, 174720, 1071360, 166320

Offset: 1

Jaroslav Krizek, Aug 08 2019

nonn

			For n = 3; a(3) = 30 because 30 is the smallest number with exactly 3 divisors d that sigma(d) is also its divisor, namely 1, 2 and 5 as sigma(1) = 1, sigma(2) = 3 and sigma(5) = 6, and all these (1, 3 and 6) are divisors of 30.

Cf. A173441, A326696, A326697, A326698.

A309253:=func; [A309253(n): n in[1..40]];

f[n_] := Count[Divisors[n], ?(Divisible[n, DivisorSigma[1, #]] &)]; m = 42; s = Table[0, {m}]; c = 0; n = 1; While[c < m, v = f[n]; If[v <=m && s[[v]] == 0, c++; s[[v]] = n]; n++]; s (* _Amiram Eldar, Aug 08 2019 *)

a(n) = my(m=1); while(sumdiv(m, d, !(m%sigma(d))) != n, m++); m; \\ Jinyuan Wang, Aug 08 2019

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

Original entry on oeis.org

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

Jaroslav Krizek, Aug 30 2019

nonn

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)

			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.

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

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

Subsequences: A008588 \ {0}, A097603.

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

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

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

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

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

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A326698 a(n) is the product of divisors d of n such that sigma(d) divides n.

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A309253 a(n) is the smallest number m with exactly n such divisors d that sigma(d) divides m.

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

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