A293433 a(n) is the number of the proper divisors of n that are Jacobsthal numbers (A001045).

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 2, 1, 1, 2, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 3, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 2, 1, 1, 3, 2, 2, 2, 2, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 4

Offset: 1

Author

Antti Karttunen, Oct 09 2017

Keywords

• nonn

Examples

			For n = 21, whose proper divisors are [1, 3, 7], both 1 and 3 are in A001045, thus a(21) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..21845

Crossrefs

Cf. A001045, A147612, A293431, A293434.

Cf. also A293435.

Programs

Mathematica

With[{s = LinearRecurrence[{1, 2}, {0, 1}, 24]}, Table[DivisorSum[n, 1 &, And[MemberQ[s, #], # != n] &], {n, 105}]] (* Michael De Vlieger, Oct 09 2017 *)

PARI

A147612aux(n,i) = if(!(n%2),n,A147612aux((n+i)/2,-i));
A147612(n) = 0^(A147612aux(n,1)*A147612aux(n,-1));
A293433(n) = sumdiv(n,d,(dA147612(d));

Python

from sympy import divisors
def A293433(n): return sum(1 for d in divisors(n,generator=True) if d(m-3).bit_length()) # Chai Wah Wu, Apr 18 2025

Formula

a(n) = Sum_{d|n, dA147612(d).

a(n) = A293431(n) - A147612(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=2} 1/A001045(n) = 1.718591611927... . - Amiram Eldar, Jul 05 2025