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A274262 Number of positive integers possessing exactly n Fibonacci representations (A000121). 1
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 (list; graph; refs; listen; history; text; internal format)
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: A068065 A278228 A028328 * A092990 A323505 A350355

Adjacent sequences: A274259 A274260 A274261 * A274263 A274264 A274265

KEYWORD

nonn

AUTHOR

Steven Finch, Jun 16 2016

STATUS

approved

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Last modified February 5 03:48 EST 2023. Contains 360082 sequences. (Running on oeis4.)