A274262 Number of positive integers possessing exactly n Fibonacci representations (A000121).

1, 2, 4, 6, 8, 12, 12, 18, 20, 24, 20, 44, 24, 36, 48, 54, 32, 76, 36, 88, 72, 60, 44, 156, 72, 72, 100, 132, 56, 208, 60, 162, 120, 96, 144, 316, 72, 108, 144, 312, 80, 312, 84, 220, 304, 132, 92, 540, 156, 280, 192, 264, 104, 460, 240, 468, 216, 168, 116, 116, 120, 180, 456, 486, 288, 520, 132, 352, 264, 624, 140

Offset

1,2

Links

Table of n, a(n) for n=1..71.

Zai-Qiao Bai and Steven R. Finch, Fibonacci and Lucas Representations, Fibonacci Quart. 54 (2016), no. 4, 319-326.

Formula

Let p, q, r be distinct primes and k be a positive integer.

If n = p^k then a(n) = 2*(p-1)*(2*p-1)^(k-1).

If n = p*q then a(n) = 6*(p-1)*(q-1).

If n = p^2*q then a(n) = 2*(p-1)*(8*p-5)*(q-1).

If n = p^3*q then a(n) = 2*(p-1)*(2*p-1)*(10*p-7)*(q-1).

If n = p^4*q then a(n) = 6*(p-1)*(2*p-1)^2*(4*p-3)*(q-1).

If n = p^2*q^2 then a(n) = 2*(p-1)*(q-1)*(26*p*q-18*p-18*q+13).

If n = p*q*r then a(n) = 26*(p-1)*(q-1)*(r-1).

Example

Let phi denote the Euler totient.
The integer p^2*q has 8 multiplicative compositions:
  (p^2*q), p^2*q, q*p^2, p*(p*q), (p*q)*p, q*p*p, p*q*p, p*p*q
from which
  a(p^2*q) = 2*(3*phi(p^2)*phi(q) + 5*phi(p)^2*phi(q))
follows immediately.

Crossrefs

Cf. A000121, A067595.

Sequence in context: A278228 A028328 A360408 * A092990 A323505 A350355

Adjacent sequences: A274259 A274260 A274261 * A274263 A274264 A274265

Keyword

nonn

Author

Steven Finch, Jun 16 2016

Status

approved