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A357171 a(n) is the number of divisors of n whose digits are in strictly increasing order (A009993).

%I A357171 #36 Jan 06 2024 09:21:37
%S A357171 1,2,2,3,2,4,2,4,3,3,1,6,2,4,4,5,2,6,2,4,3,2,2,8,3,4,4,6,2,6,1,5,2,4,
%T A357171 4,9,2,4,4,5,1,6,1,3,6,4,2,10,3,4,3,5,1,7,2,8,4,4,2,8,1,2,4,5,3,4,2,6,
%U A357171 4,6,1,11,1,3,5,5,2,8,2,6,4,2,1,9,3,2,3,4,2,9,3,5,2,3,3,10,1,5,3,5
%N A357171 a(n) is the number of divisors of n whose digits are in strictly increasing order (A009993).
%C A357171 As A009993 is finite with 512 terms, a(n) is bounded with a(n) <= 511 and not 512, since A009993(1) = 0.
%F A357171 G.f.: Sum_{n in A009993} x^n/(1-x^n). - _Robert Israel_, Sep 16 2022
%F A357171 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n=2..512} 1/A009993(n) = 4.47614714667538759358... (this is a rational number whose numerator and denominator have 1037 and 1036 digits, respectively). - _Amiram Eldar_, Jan 06 2024
%e A357171 22 has 4 divisors {1, 2, 11, 22} of which two have decimal digits that are not in strictly increasing order: {11, 22}, hence a(22) = 4-2 = 2.
%e A357171 52 has divisors {1, 2, 4, 13, 26, 52} and a(52) = 5 of them have decimal digits that are in strictly increasing order (all except 52 itself).
%p A357171 f:= proc(n) local d,L,i,t;
%p A357171   t:= 0;
%p A357171   for d in numtheory:-divisors(n) do
%p A357171     L:= convert(d,base,10);
%p A357171     if `and`(seq(L[i]>L[i+1],i=1..nops(L)-1)) then t:= t+1 fi
%p A357171   od;
%p A357171   t
%p A357171 end proc:
%p A357171 map(f, [$1..100]); # _Robert Israel_, Sep 16 2022
%t A357171 a[n_] := DivisorSum[n, 1 &, Less @@ IntegerDigits[#] &]; Array[a, 100] (* _Amiram Eldar_, Sep 16 2022 *)
%o A357171 (PARI) isok(d) = Set(d=digits(d)) == d; \\ A009993
%o A357171 a(n) = sumdiv(n, d, isok(d)); \\ _Michel Marcus_, Sep 16 2022
%o A357171 (Python)
%o A357171 from sympy import divisors
%o A357171 def c(n): s = str(n); return s == "".join(sorted(set(s)))
%o A357171 def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
%o A357171 print([a(n) for n in range(1, 101)]) # _Michael S. Branicky_, Sep 16 2022
%Y A357171 Cf. A009993, A357172, A357173, A160218.
%Y A357171 Similar: A087990 (palindromic), A355302 (undulating), A355593 (alternating).
%K A357171 nonn,base
%O A357171 1,2
%A A357171 _Bernard Schott_, Sep 16 2022