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%I #12 Jan 01 2023 12:36:41
%S 1,2,4,6,8,12,12,18,20,24,20,44,24,36,48,54,32,76,36,88,72,60,44,156,
%T 72,72,100,132,56,208,60,162,120,96,144,316,72,108,144,312,80,312,84,
%U 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
%N Number of positive integers possessing exactly n Fibonacci representations (A000121).
%H Zai-Qiao Bai and Steven R. Finch, <a href="https://www.fq.math.ca/Papers1/54-4/BaiFinch09122016.pdf">Fibonacci and Lucas Representations</a>, Fibonacci Quart. 54 (2016), no. 4, 319-326.
%F Let p, q, r be distinct primes and k be a positive integer.
%F If n = p^k then a(n) = 2*(p-1)*(2*p-1)^(k-1).
%F If n = p*q then a(n) = 6*(p-1)*(q-1).
%F If n = p^2*q then a(n) = 2*(p-1)*(8*p-5)*(q-1).
%F If n = p^3*q then a(n) = 2*(p-1)*(2*p-1)*(10*p-7)*(q-1).
%F If n = p^4*q then a(n) = 6*(p-1)*(2*p-1)^2*(4*p-3)*(q-1).
%F If n = p^2*q^2 then a(n) = 2*(p-1)*(q-1)*(26*p*q-18*p-18*q+13).
%F If n = p*q*r then a(n) = 26*(p-1)*(q-1)*(r-1).
%e Let phi denote the Euler totient.
%e The integer p^2*q has 8 multiplicative compositions:
%e (p^2*q), p^2*q, q*p^2, p*(p*q), (p*q)*p, q*p*p, p*q*p, p*p*q
%e from which
%e a(p^2*q) = 2*(3*phi(p^2)*phi(q) + 5*phi(p)^2*phi(q))
%e follows immediately.
%Y Cf. A000121, A067595.
%K nonn
%O 1,2
%A _Steven Finch_, Jun 16 2016
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