A337326 a(n) is the smallest number with n divisors d such that sigma(d) / tau(d) is an integer.

1, 3, 6, 15, 45, 30, 42, 60, 132, 264, 270, 378, 594, 210, 462, 780, 1050, 420, 924, 660, 2100, 840, 3060, 1848, 3300, 1890, 2970, 2520, 9702, 2310, 5544, 3780, 11592, 8316, 18216, 5460, 5940, 7980, 16830, 7140, 11550, 4620, 21252, 10920, 23760, 22440, 49500

Offset: 1

Author

Jaroslav Krizek, Oct 20 2020

Keywords

• nonn

Comments

a(n) is the smallest number m with n arithmetic divisors d (terms of A003601).

See A338170, A338171 and A338172 for number, sum and product of such divisors for n>=1.

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A003601 (arithmetic numbers).

Cf. A334421 (smallest number with n divisors d such that sigma(d)/tau(d) is an integer for all divisors).

Programs

Magma

[Min([m: m in[1..10^5] | #[d: d in Divisors(m) | IsIntegral(&+Divisors(d) / #Divisors(d))] eq n]): n in [1..30]];

Mathematica

f[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &]; m = 50; s = Table[0, {m}]; c = 0; n = 1; While[c < m, If[(i = f[n]) <= m && s[[i]] == 0, c++; s[[i]] = n]; n++]; s (* Amiram Eldar, Oct 21 2020 *)

PARI

isok(m, n) = sumdiv(m, d, !(sigma(d) % numdiv(d))) == n;
a(n) = my(m=1); while(!isok(m,n), m++); m; \\ Michel Marcus, Oct 21 2020

Formula

a(3) = 6 because number 6 is the smallest number with 3 such divisors (1, 3 and 6): sigma(1) / tau(1) = 1 / 1 = 1; sigma(3) / tau(3) = 4 / 2 = 2; sigma(6) / tau(6) = 12 / 4 = 3.