A304096 Number of Lucas numbers larger than 3 (4, 7, 11, 18, ...) that divide n.

0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 1, 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, 2, 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 divisors d of n that are of the form d = A000045(k-1) + A000045(k+1), for k >= 3.

Examples

			The divisors of 4 are 1, 2 and 4. Of these only 4 is a Lucas number larger than 3, thus a(4) = 1.
The divisors of 28 are 1, 2, 4, 7, 14 and 28. 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. A000032, A000045, A000204, A093540, A102460, A304092, A304094, A304095, A304104.

Cf. also A005086, A300837.

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)); };
A304096(n) = sumdiv(n,d,(d>3)*A102460(d));

Formula

a(n) = Sum_{d|n, d>3} A102460(d).

a(n) = A304094(n) - A079978(n) - 1.

a(n) = A304092(n) - A059841(n) - A079978(n) - 1.

a(n) = A007949(A304104(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540 - 4/3 = 0.629524... . - Amiram Eldar, Dec 31 2023