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Integers whose number of divisors that are Zuckerman numbers sets a new record.
2

%I #33 Jul 19 2021 01:23:40

%S 1,2,4,6,12,24,72,144,360,432,1080,2016,2160,6048,8064,15120,24192,

%T 48384,88704,120960,241920,266112,532224,1064448,1862784,2661120,

%U 3725568,5322240,7451136,10450944,19160064,20901888,28740096,38320128,57480192,99283968,114960384

%N Integers whose number of divisors that are Zuckerman numbers sets a new record.

%C A Zuckerman number is a number that is divisible by the product of its digits (A007602).

%C The terms in this sequence are not necessarily Zuckerman numbers. For example a(7) = 72 has product of digits = 14 and 72/14 = 36/7 = 5.142...

%C The first seven terms are the first seven terms of A087997, then A087997(8) = 66 while a(8) = 144.

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

%e The 8 divisors of 24 are all Zuckerman numbers, and also, 24 is the smallest integer that has at least 8 divisors that are Zuckerman numbers, hence 24 is a term.

%t zuckQ[n_] := (prod = Times @@ IntegerDigits[n]) > 0 && Divisible[n, prod]; s[n_] := DivisorSum[n, 1 &, zuckQ[#] &]; smax = 0; seq = {}; Do[s1 = s[n]; If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 10^5}]; seq (* _Amiram Eldar_, Jan 14 2021 *)

%o (PARI) isokz(n) = iferr(!(n % vecprod(digits(n))), E, 0); \\ A007602

%o lista(nn) = {my(m=0); for (n=1, nn, my(x = sumdiv(n, d, isokz(d));); if (x > m, m = x; print1(n, ", ")););} \\ _Michel Marcus_, Jan 15 2021

%Y Cf. A007602, A335037, A337941.

%Y Subsequence of A335038.

%Y Similar for palindromes (A093036), repdigits (A340548), repunits (A340549), Niven numbers (A340637).

%K nonn,base

%O 1,2

%A _Bernard Schott_, Jan 14 2021

%E More terms from _David A. Corneth_ and _Amiram Eldar_, Jan 15 2021