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A115063
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Natural numbers of the form p^F(n_p)*q^F(n_q)*r^F(n_r)*...*z^F(n_z), where p,q,r,... are distinct primes and F(n) is a Fibonacci number.
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7
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67
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OFFSET
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1,2
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COMMENTS
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The complementary sequence is 16, 48, 64, 80, 81, 112, 128, 144, 162, 176, 192, 208, 240, 272, 304, 320, 324, 336, 368, 384, 400, ... - R. J. Mathar, Apr 22 2010
Let h_k be the density of the subsequence of A115063 of numbers whose prime power factorization has the form Product_i p_i^e_i where the e_i all squares <= k^2. Then for every k>1 there exists eps_k>0 such that for any x from the interval (h_k-eps_k, h_k) there is no a sequence S of positive integers such that x is the density of numbers whose prime power factorization has the form Product_i p_i^e_i where the e_i are all in S. For a proof, see [Shevelev], the second link. - Vladimir Shevelev, Nov 17 2015
Numbers whose sets of unitary divisors (A077610) and Zeckendorf-infinitary divisors (see A318465) coincide. Also, numbers whose sets of unitary divisors and dual-Zeckendorf-infinitary divisors (see A331109) coincide. - Amiram Eldar, Aug 09 2024
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LINKS
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FORMULA
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Sum_{i<=x, i is in A115063} 1 = h*x + O(sqrt(x)*log x*e^(c*sqrt(log x)/(log(log x))), where c = 4*sqrt(2.4/log 2) = 7.44308... and h = Product_{prime p}(1 + Sum_{i>=2} (u(i)-u(i-1))/p^i) = 0.944335905... where u(n) is the characteristic function of sequence A000045. The calculations of h over the formula were done independently by Juan Arias-de-Reyna and Peter J. C. Moses.
For a proof of the formula, see [Shevelev], the first link. - Vladimir Shevelev, Nov 17 2015
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EXAMPLE
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12 is a term, since 12=2^2*3^1 and the exponents 2 and 1 are terms of Fibonacci sequence (A000045). - Vladimir Shevelev, Nov 15 2015
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MATHEMATICA
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fibQ[n_] := IntegerQ @ Sqrt[5 n^2 - 4] || IntegerQ @ Sqrt[5 n^2 + 4]; aQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], fibQ]; Select[Range[100], aQ] (* Amiram Eldar, Oct 06 2019 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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