cp's OEIS Frontend

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

%I A293433 #16 Jul 05 2025 09:54:59
%S A293433 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,
%T A293433 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,
%U A293433 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
%N A293433 a(n) is the number of the proper divisors of n that are Jacobsthal numbers (A001045).
%H A293433 Antti Karttunen, <a href="/A293433/b293433.txt">Table of n, a(n) for n = 1..21845</a>
%F A293433 a(n) = Sum_{d|n, d<n} A147612(d).
%F A293433 a(n) = A293431(n) - A147612(n).
%F A293433 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
%e A293433 For n = 21, whose proper divisors are [1, 3, 7], both 1 and 3 are in A001045, thus a(21) = 2.
%t A293433 With[{s = LinearRecurrence[{1, 2}, {0, 1}, 24]}, Table[DivisorSum[n, 1 &, And[MemberQ[s, #], # != n] &], {n, 105}]] (* _Michael De Vlieger_, Oct 09 2017 *)
%o A293433 (PARI)
%o A293433 A147612aux(n,i) = if(!(n%2),n,A147612aux((n+i)/2,-i));
%o A293433 A147612(n) = 0^(A147612aux(n,1)*A147612aux(n,-1));
%o A293433 A293433(n) = sumdiv(n,d,(d<n)*A147612(d));
%o A293433 (Python)
%o A293433 from sympy import divisors
%o A293433 def A293433(n): return sum(1 for d in divisors(n,generator=True) if d<n and (m:=3*d+1).bit_length()>(m-3).bit_length()) # _Chai Wah Wu_, Apr 18 2025
%Y A293433 Cf. A001045, A147612, A293431, A293434.
%Y A293433 Cf. also A293435.
%K A293433 nonn
%O A293433 1,6
%A A293433 _Antti Karttunen_, Oct 09 2017