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a(n) is the smallest number m with exactly n divisors that are Zuckerman numbers, or -1 if there is no such m.
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%I #17 May 06 2022 13:13:51

%S 1,2,4,6,18,12,84,24,168,72,144,360,432,1080,2016,2160,6048,8064,

%T 15120,34272,24192,60480,48384,88704,120960,354816,241920,483840,

%U 665280,266112,798336,532224,1596672,1064448,1862784,2661120,3725568,5322240,10644480,7451136

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

%C Inspired by A333456.

%C 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.

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

%C Conjecture: m always exists.

%C 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.

%H Giovanni Resta, <a href="http://www.numbersaplenty.com/set/Zuckerman_number/">Zuckerman numbers</a>, Numbers Aplenty.

%e 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.

%t 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 *)

%o (PARI) iszu(n) = my(p=vecprod(digits(n))); p && !(n % p);

%o a(n) = {my(k=1); while (sumdiv(k, d, iszu(d)) !=n, k++); k;} \\ _Michel Marcus_, Jun 03 2020

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

%K nonn,base

%O 1,2

%A _Bernard Schott_, Jun 03 2020

%E More terms from _Amiram Eldar_, Jun 03 2020

%E Edited, added escape clause. - _N. J. A. Sloane_, Jun 04 2020