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

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, 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, 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

Offset

1,2

Comments

As A009995 is finite with 1023 terms, a(n) is bounded with a(n) <= 1022 and not 1023, since A009995(1) = 0.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

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

Example

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.
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.

Maple

f:= proc(n) local L;
   if n < 10 then return true fi;
   L:= convert(n, base, 10);
   andmap(type, L[2..-1]-L[1..-2], positive)
end proc:
g:= n -> nops(select(f, numtheory:-divisors(n))):
map(g, [$1..100]); # Robert Israel, Oct 31 2022

Mathematica

a[n_] := DivisorSum[n, 1 &, Max @ Differences @ IntegerDigits[#] < 0 &]; Array[a, 100] (* Amiram Eldar, Oct 29 2022 *)

Prog

(PARI) a(n) = sumdiv(n, d, my(dd=digits(d)); vecsort(dd, , 12) == dd); \\ Michel Marcus, Oct 30 2022
(Python)
from sympy import divisors
def c(n): s = str(n); return all(s[i+1] < s[i] for i in range(len(s)-1))
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, Feb 12 2024

Crossrefs

Cf. A009995, A190219, A358100, A358101.

Similar: A086971 (semiprimes), A087990 (palindromic), A355593 (alternating), A357171 (increasing order).

Sequence in context: A356069 A179940 A138707 * A095048 A332268 A355593

Adjacent sequences: A358096 A358097 A358098 * A358100 A358101 A358102

Keyword

nonn,base

Author

Bernard Schott, Oct 29 2022

Status

approved