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

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, 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, 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

Offset

1,2

Comments

As A009993 is finite with 512 terms, a(n) is bounded with a(n) <= 511 and not 512, since A009993(1) = 0.

Links

Table of n, a(n) for n=1..100.

Formula

G.f.: Sum_{n in A009993} x^n/(1-x^n). - Robert Israel, Sep 16 2022

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

Example

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.
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).

Maple

f:= proc(n) local d, L, i, t;
  t:= 0;
  for d in numtheory:-divisors(n) do
    L:= convert(d, base, 10);
    if `and`(seq(L[i]>L[i+1], i=1..nops(L)-1)) then t:= t+1 fi
  od;
  t
end proc:
map(f, [$1..100]); # Robert Israel, Sep 16 2022

Mathematica

a[n_] := DivisorSum[n, 1 &, Less @@ IntegerDigits[#] &]; Array[a, 100] (* Amiram Eldar, Sep 16 2022 *)

Prog

(PARI) isok(d) = Set(d=digits(d)) == d; \\ A009993
a(n) = sumdiv(n, d, isok(d)); \\ Michel Marcus, Sep 16 2022
(Python)
from sympy import divisors
def c(n): s = str(n); return s == "".join(sorted(set(s)))
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Sep 16 2022

Crossrefs

Cf. A009993, A357172, A357173, A160218.

Similar: A087990 (palindromic), A355302 (undulating), A355593 (alternating).

Sequence in context: A306509 A083867 A076888 * A355698 A087990 A335037

Adjacent sequences: A357168 A357169 A357170 * A357172 A357173 A357174

Keyword

nonn,base

Author

Bernard Schott, Sep 16 2022

Status

approved