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A288069 Quotients obtained when the Zuckerman numbers are divided by the product of their digits.

%I A288069 #43 Apr 12 2025 11:56:12
%S A288069 1,1,1,1,1,1,1,1,1,11,6,3,3,2,111,56,23,8,22,9,9,5,53,18,14,52,21,4,
%T A288069 18,51,13,8,7,17,1111,556,371,223,186,377,28,37,19,303,12,437,74,28,
%U A288069 59,9,49,528,67,93,27,1037,174,22,151,13,184,29,514,66,46
%N A288069 Quotients obtained when the Zuckerman numbers are divided by the product of their digits.
%C A288069 The Zuckerman numbers (A007602) are the numbers that are divisible by the product of their digits.
%C A288069 Question: Is A067251 a subsequence? No, it appears in A056770 that not all integers other than multiples of 10 can be obtained as quotient, such as 15, 16, 24, 25, 26, 32, .... (see A342941).
%C A288069 The limit of the sequence is infinite: for any x, there is some N such that, for all n > N, a(n) > x. Proof: a Zuckerman number with d digits is at least 10^(d-1) and has a digit product at most 9^d and so has a quotient at least 10^(d-1)/9^d which goes to infinity with d. - _Charles R Greathouse IV_, Jun 05 2017
%C A288069 The repunits A002275 are a subsequence. - _Peter Schorn_, Apr 05 2025
%H A288069 Charles R Greathouse IV, <a href="/A288069/b288069.txt">Table of n, a(n) for n = 1..10000</a>
%e A288069 a(11) = 12/(1*2) = 6; a(13) = 24/(2*4) = 3.
%p A288069 f:= proc(n) local L,p;
%p A288069    p:= convert(convert(n,base,10),`*`);
%p A288069    if p > 0 then
%p A288069      if n mod p = 0 then return n/p fi
%p A288069    fi
%p A288069 end proc:
%p A288069 map(f, [$1..10^4]); # _Robert Israel_, Jun 05 2017
%t A288069 Select[Table[n/Max[Times@@IntegerDigits[n],Pi/100],{n,5000}],IntegerQ] (* _Harvey P. Dale_, Aug 16 2021 *)
%Y A288069 Cf. A007602, A067251, A056770, A342593, A342941, A002275.
%K A288069 nonn,base
%O A288069 1,10
%A A288069 _Bernard Schott_, Jun 05 2017