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A383592 Positive integers k divisible by all positive integers whose decimal expansion appears as a substring of k.

%I A383592 #16 May 13 2025 11:33:49
%S A383592 1,2,3,4,5,6,7,8,9,10,11,12,15,20,22,24,30,33,36,40,44,48,50,55,60,66,
%T A383592 70,77,80,88,90,99,100,110,120,150,200,210,220,240,250,300,330,360,
%U A383592 400,420,440,480,500,510,520,550,600,630,660,700,770,800,840,880
%N A383592 Positive integers k divisible by all positive integers whose decimal expansion appears as a substring of k.
%C A383592 This sequence is infinite as ten times a term is also a term.
%C A383592 All terms are of the form A037124(k) or A037124(k) + d where k > 0 and d divides A037124(k) while having strictly less decimal digits as A037124(k).
%C A383592 Empirically, all terms have either one or two nonzero decimal digits.
%H A383592 Rémy Sigrist, <a href="/A383592/b383592.txt">Table of n, a(n) for n = 1..10314</a>
%H A383592 Rémy Sigrist, <a href="/A383592/a383592.gp.txt">PARI program</a>
%e A383592 The number 240 is divisible by 2, 24, 240, 4 and 40, so 240 belongs to this sequence.
%t A383592 Select[Range[880],AllTrue[#/Select[FromDigits/@Subsequences[IntegerDigits[#]],#>0&],IntegerQ]&] (* _James C. McMahon_, May 13 2025 *)
%o A383592 (PARI) is(n, base = 10) = {
%o A383592     my (d = digits(n, base));
%o A383592     for (i = 1, #d,
%o A383592         if (d[i],
%o A383592             for (j = i, #d,
%o A383592                 if (n % fromdigits(d[i..j], base),
%o A383592                     return (0);););););
%o A383592     return (1); }
%o A383592 (PARI) \\ See Links section.
%o A383592 (Python)
%o A383592 def ok(n):
%o A383592     s = str(n)
%o A383592     subs = (s[i:j] for i in range(len(s)) for j in range(i+1, len(s)+1) if s[i]!='0')
%o A383592     return n and all(n%v == 0 for ss in subs if (v:=int(ss)) > 0)
%o A383592 print([k for k in range(1000) if ok(k)]) # _Michael S. Branicky_, May 09 2025
%Y A383592 Cf. A037124, A078546, A175381 (binary variant), A178157, A218978.
%K A383592 nonn,base,easy
%O A383592 1,2
%A A383592 _Rémy Sigrist_, May 01 2025