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

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

Author

Jaroslav Krizek, Oct 24 2020

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

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

Crossrefs

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

Cf. A325469, A325470, A325471, A337691.

Programs

Magma

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

Mathematica

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 *)

PARI

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

Extensions

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