A357171 - OEIS

%I #36 Jan 06 2024 09:21:37

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

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

%F G.f.: Sum_{n in A009993} x^n/(1-x^n). - _Robert Israel_, Sep 16 2022

%F 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 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 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 f:= proc(n) local d,L,i,t;

%p   t:= 0;

%p   for d in numtheory:-divisors(n) do

%p     L:= convert(d,base,10);

%p     if `and`(seq(L[i]>L[i+1],i=1..nops(L)-1)) then t:= t+1 fi

%p   od;

%p   t

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Sep 16 2022

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

%o (PARI) isok(d) = Set(d=digits(d)) == d; \\ A009993

%o a(n) = sumdiv(n, d, isok(d)); \\ _Michel Marcus_, Sep 16 2022

%o (Python)

%o from sympy import divisors

%o def c(n): s = str(n); return s == "".join(sorted(set(s)))

%o def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))

%o print([a(n) for n in range(1, 101)]) # _Michael S. Branicky_, Sep 16 2022

%Y Cf. A009993, A357172, A357173, A160218.

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

%K nonn,base

%O 1,2

%A _Bernard Schott_, Sep 16 2022