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
Examples
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.
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
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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
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Mathematica
a[n_] := DivisorSum[n, 1 &, Max @ Differences @ IntegerDigits[#] < 0 &]; Array[a, 100] (* Amiram Eldar, Oct 29 2022 *)
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PARI
a(n) = sumdiv(n, d, my(dd=digits(d)); vecsort(dd, ,12) == dd); \\ Michel Marcus, Oct 30 2022
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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
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
Comments