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

1,28

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.

Links

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

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

Example

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.

Prog

(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));

Crossrefs

Cf. A000032, A000045, A000204, A093540, A102460, A304092, A304094, A304095, A304104.

Cf. also A005086, A300837.

Sequence in context: A345374 A056929 A374444 * A151692 A280829 A303942

Adjacent sequences: A304093 A304094 A304095 * A304097 A304098 A304099

Keyword

nonn

Author

Antti Karttunen, May 13 2018

Status

approved