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A357172 a(n) is the smallest integer that has exactly n divisors whose decimal digits are in strictly increasing order.

%I A357172 #32 Sep 17 2022 14:11:36
%S A357172 1,2,4,6,16,12,54,24,36,48,72,180,144,360,336,468,504,936,1008,1512,
%T A357172 2520,3024,5040,6552,7560,22680,13104,19656,49140,105840,39312,78624,
%U A357172 98280,248976,334152,196560,393120,668304,1244880,1670760,1867320,4520880,3341520,3734640
%N A357172 a(n) is the smallest integer that has exactly n divisors whose decimal digits are in strictly increasing order.
%C A357172 This sequence is finite since A009993 is finite with 511 nonzero terms, hence the last term is a(511) = lcm of the 511 nonzero terms of A009993.
%C A357172 a(511) = 8222356410...6120992000 and has 1036 digits. - _Michael S. Branicky_, Sep 16 2022
%e A357172 For n=7, the divisors of 54 are {1, 2, 3, 6, 9, 18, 27, 54} of which 7 have their digits in strictly increasing order (all except 54). No integer < 54 has 7 such divisors, so a(7) = 54.
%t A357172 s[n_] := DivisorSum[n, 1 &, Less @@ IntegerDigits[#] &]; seq[len_, nmax_] := Module[{v = Table[0, {len}], n = 1, c = 0, i}, While[c < len && n < nmax, i = s[n]; If[i <= len && v[[i]] == 0, v[[i]] = n; c++]; n++]; v]; seq[25, 10^4] (* _Amiram Eldar_, Sep 16 2022 *)
%o A357172 (PARI) isok(d) = Set(d=digits(d)) == d; \\ A009993
%o A357172 f(n) = sumdiv(n, d, isok(d)); \\ A357171
%o A357172 a(n) = my(k=1); while (f(k) !=n, k++); k; \\ _Michel Marcus_, Sep 16 2022
%o A357172 (Python)
%o A357172 from sympy import divisors
%o A357172 from itertools import count, islice
%o A357172 def c(n): s = str(n); return s == "".join(sorted(set(s)))
%o A357172 def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
%o A357172 def agen():
%o A357172     n, adict = 1, dict()
%o A357172     for k in count(1):
%o A357172         fk = f(k)
%o A357172         if fk not in adict: adict[fk] = k
%o A357172         while n in adict: yield adict[n]; n += 1
%o A357172 print(list(islice(agen(), 37))) # _Michael S. Branicky_, Sep 16 2022
%Y A357172 Cf. A009993, A357171, A357173, A160218.
%Y A357172 Similar: A087997 (palindromic), A355303 (undulating), A355594 (alternating).
%K A357172 nonn,base,fini
%O A357172 1,2
%A A357172 _Bernard Schott_, Sep 16 2022
%E A357172 More terms from _Amiram Eldar_, Sep 16 2022