A325469 -id:A325469 - cp's OEIS frontend

A325470 a(n) is the sum of divisors d of n such that d divides sigma(d).

1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 29, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 29, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 35, 1, 1

Offset: 1

Jaroslav Krizek, Aug 16 2019

nonn

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

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

Cf. A000203, A097603, A325469, A325471.

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

a[n_] := DivisorSum[n, # &, Divisible[DivisorSigma[1, #], #] &];  Array[a, 100] (* Amiram Eldar, Aug 17 2019 *)

a(n)={sumdiv(n, d, if(sigma(d) % d == 0, d))} \\ Andrew Howroyd, Aug 16 2019

a(A097603(n)) > 1.

A325471 a(n) is the product of divisors d of n such that d divides sigma(d).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 28, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 28, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 168, 1, 1

Offset: 1

Jaroslav Krizek, Aug 16 2019

nonn

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

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000203, A097603, A325469, A325470.

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

a[n_] := Times @@ Select[Divisors[n], Divisible[DivisorSigma[1, #], #] &]; Array[a, 100] (* Amiram Eldar, Aug 17 2019 *)

a(n)={vecprod([d | d<-divisors(n), sigma(d) % d==0])} \\ Andrew Howroyd, Aug 16 2019

a(A097603(n)) > 1.

A338405 a(n) is the smallest number with exactly n divisors d such that sigma(d)/d is an integer.

Original entry on oeis.org

1, 6, 84, 672, 3360, 30240, 393120, 12186720, 164989440, 14024102400, 2144862720, 182313331200, 5705334835200, 96990692198400, 187409525022720, 9602078527641600, 124627334140108800, 2118664680381849600, 19067982123436646400, 209747803357803110400, 3985208263798259097600, 63343836614056539340800, 401177631889024749158400, 1203532895667074247475200

Offset: 1

Jaroslav Krizek, Oct 24 2020

nonn

a(n) is the smallest number with n multiply-perfect divisors.

Number 1 is only number m such that sigma(d) / d is an integer for all divisors d.

			a(3) = 84 because 84 with divisors 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84 is the smallest number with 3 multiply-perfect divisors (1, 6 and 28): sigma(1)/1 = 1, sigma(6)/6  = 2, sigma(28)/28  = 2.

Max Alekseyev, Table of n, a(n) for n = 1..51
David A. Corneth, PARI program

Cf. A000203 (sigma), A007691 (multiply-perfect numbers).

Cf. A325469, A325470, A325471, A337691.

[Min([m: m in[1..10^5] | #[d: d in Divisors(m) | IsIntegral(&+Divisors(d) / d)] eq n]): n in [1..6]]

f[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[1, #], #] &]; m = 7; s = Table[0, {m}]; c = 0; n = 1; While[c < m, i = f[n]; If[i <= m && s[[i]] == 0, c++; s[[i]] = n]; n++]; s (* Amiram Eldar, Oct 25 2020 *)

a(n) = {my(m=1); while (sumdiv(m, d, !(sigma(d)%d)) != n, m++); m;} \\ Michel Marcus, Oct 25 2020

a(8) from Michel Marcus, Oct 25 2020
a(9)-a(13) from Jinyuan Wang, Oct 31 2020
Name clarified by Chai Wah Wu, Nov 01 2020
a(14)-a(20) from David A. Corneth, Nov 02 2020
Terms a(21) onward from Max Alekseyev, Feb 21 2024

cp's OEIS Frontend

A325470 a(n) is the sum of divisors d of n such that d divides sigma(d).

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A325471 a(n) is the product of divisors d of n such that d divides sigma(d).

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Author

Keywords

Examples

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Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A338405 a(n) is the smallest number with exactly n divisors d such that sigma(d)/d is an integer.

Views

Author

Keywords

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Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

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