A304095 a(n) is the number of the proper divisors of n that are Lucas numbers larger than 3 (4, 7, 11, 18, ...).

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 2, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 1, 1

Offset: 1

Author

Antti Karttunen, May 13 2018

Keywords

• nonn

Comments

a(n) is the number of the proper divisors d of n that are of the form d = A000045(k-1) + A000045(k+1), for k >= 3.

Examples

			The proper divisors of 28 are 1, 2, 4, 7 and 14. Of these 4 and 7 are Lucas numbers (A000032) larger than 3, thus a(28) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000045, A000204, A093540, A102460, A304091, A304093, A304096, A304102.

Cf. also A300836.

Programs

PARI

A102460(n) = { my(u1=1,u2=3,old_u1); if(n<=2,sign(n),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304095(n) = sumdiv(n,d,(d>3)*(dA102460(d));

Formula

a(n) = Sum_{d|n, d>3, dA102460(d).

a(n) = A007949(A304102(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540 - 4/3 = 0.629524... . - Amiram Eldar, Jul 05 2025