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A358099 a(n) is the number of divisors of n whose digits are in strictly decreasing order (A009995).

%I A358099 #30 Feb 12 2024 17:23:25
%S A358099 1,2,2,3,2,4,2,4,3,4,1,5,1,3,3,4,1,5,1,6,4,2,1,6,2,2,3,4,1,7,2,5,2,2,
%T A358099 3,6,1,2,2,8,2,7,2,3,4,2,1,6,2,5,3,4,2,6,2,5,2,2,1,10,2,4,6,6,3,4,1,3,
%U A358099 2,6,2,8,2,3,4,4,2,4,1,9,4,4,2,9,3,4,3,4,1,9,3,4,4,3,3,8,2,4,3,7
%N A358099 a(n) is the number of divisors of n whose digits are in strictly decreasing order (A009995).
%C A358099 As A009995 is finite with 1023 terms, a(n) is bounded with a(n) <= 1022 and not 1023, since A009995(1) = 0.
%H A358099 Robert Israel, <a href="/A358099/b358099.txt">Table of n, a(n) for n = 1..10000</a>
%F A358099 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n=2..1023} 1/A009995(n) = 3.89840673699905364734... (this is a rational number whose numerator and denominator have 1292 and 1291 digits, respectively). - _Amiram Eldar_, Jan 06 2024
%e A358099 22 has 4 divisors {1, 2, 11, 22} of which two have decimal digits that are not in strictly decreasing order: {11, 22}, hence a(22) = 4-2 = 2.
%e A358099 52 has 6 divisors {1, 2, 4, 13, 26, 52} of which four have decimal digits that are in strictly decreasing order {1, 2, 4, 52}, hence a(52) = 4.
%p A358099 f:= proc(n) local L;
%p A358099    if n < 10 then return true fi;
%p A358099    L:= convert(n,base,10);
%p A358099    andmap(type,L[2..-1]-L[1..-2],positive)
%p A358099 end proc:
%p A358099 g:= n -> nops(select(f,numtheory:-divisors(n))):
%p A358099 map(g, [$1..100]); # _Robert Israel_, Oct 31 2022
%t A358099 a[n_] := DivisorSum[n, 1 &, Max @ Differences @ IntegerDigits[#] < 0 &]; Array[a, 100] (* _Amiram Eldar_, Oct 29 2022 *)
%o A358099 (PARI) a(n) = sumdiv(n, d, my(dd=digits(d)); vecsort(dd, ,12) == dd); \\ _Michel Marcus_, Oct 30 2022
%o A358099 (Python)
%o A358099 from sympy import divisors
%o A358099 def c(n): s = str(n); return all(s[i+1] < s[i] for i in range(len(s)-1))
%o A358099 def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
%o A358099 print([a(n) for n in range(1, 101)]) # _Michael S. Branicky_, Feb 12 2024
%Y A358099 Cf. A009995, A190219, A358100, A358101.
%Y A358099 Similar: A086971 (semiprimes), A087990 (palindromic), A355593 (alternating), A357171 (increasing order).
%K A358099 nonn,base
%O A358099 1,2
%A A358099 _Bernard Schott_, Oct 29 2022