A335038 a(n) is the smallest number m with exactly n divisors that are Zuckerman numbers, or -1 if there is no such m.

1, 2, 4, 6, 18, 12, 84, 24, 168, 72, 144, 360, 432, 1080, 2016, 2160, 6048, 8064, 15120, 34272, 24192, 60480, 48384, 88704, 120960, 354816, 241920, 483840, 665280, 266112, 798336, 532224, 1596672, 1064448, 1862784, 2661120, 3725568, 5322240, 10644480, 7451136

Offset: 1

Author

Bernard Schott, Jun 03 2020

Keywords

• nonn
• base

Comments

Inspired by A333456.

A Zuckerman number (A007602) is a number that is divisible by the product of its digits; e.g., 24 is a Zuckerman number because it is divisible by 2*4=8.

The divisors 1 and m (if m is itself a Zuckerman number) are included.

Conjecture: m always exists.

Not all terms in the sequence are Zuckerman numbers. For example a(7) = 84 has product of digits = 32 and 84/32 = 21/8 = 2.625.

Examples

			Of the six divisors of 18, five are Zuckerman numbers: 1, 2, 3, 6 and 9, and there is no smaller number with five Zuckerman divisors, hence a(5) = 18.
		

Links

• Giovanni Resta, Zuckerman numbers, Numbers Aplenty.

Crossrefs

Cf. A007602, A335037, A333456 (similar, with Niven divisors).

Programs

Mathematica

zuckQ[n_] := (prodig = Times @@ IntegerDigits[n]) > 0 && Divisible[n, prodig]; numDiv[n_] := Count[Divisors[n], ?(zuckQ[#] &)]; mx = 50; n = 1; c = 0; v = Table[0, {mx}]; While[c < mx, i = numDiv[n]; If[i <= mx && v[[i]] == 0, c++; v[[i]] = n]; n++]; v (* _Amiram Eldar, Jun 03 2020 *)

PARI

iszu(n) = my(p=vecprod(digits(n))); p && !(n % p);
a(n) = {my(k=1); while (sumdiv(k, d, iszu(d)) !=n, k++); k;} \\ Michel Marcus, Jun 03 2020

Extensions

More terms from Amiram Eldar, Jun 03 2020

Edited, added escape clause. - N. J. A. Sloane, Jun 04 2020