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A079598
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a(n) = 2^(4n+1) - 2^(2n).
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4
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1, 28, 496, 8128, 130816, 2096128, 33550336, 536854528, 8589869056, 137438691328, 2199022206976, 35184367894528, 562949936644096, 9007199187632128, 144115187807420416, 2305843008139952128
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OFFSET
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0,2
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COMMENTS
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A sequence rich in perfect numbers, since according to the Euclid-Euler theorem all even perfect numbers are of the form 2^(k-1)*(2^k - 1) and in this sequence k = 2*n + 1.
Let G be a sequence satisfying G(i) = 2*G(i-1) + 8*G(i-2) for arbitrary integers i and without regard to the initial values of G. Then a(n) = (G(i+4*n+2) - G(i)*8^(2*n+1))/(2*G(i+2*n+1)) as long as G(i+2*n+1) != 0. - Klaus Purath, Oct 22 2020
In the binary system, the elements of the sequence consist of a total of 4*n+1 bits starting with 2*n+1 ones followed by 2*n zeros. - Martin Renner, Mar 22 2022
Michael Stifel and Petrus Bungus erroneously claimed that for n > 0 all these numbers were perfect. See links: Bungus, Hassler, and Stifel. - Stefano Spezia, Jul 28 2024
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LINKS
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Uwe Hassler, Perfect Numbers, Euleriana: 3(2), pp.176-185, (2023). See pp. 177-178.
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FORMULA
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a(n+1) = 16*a(n) + 12*2^(2n).
a(n) = Sum_{k=1..2^n} (2*k-1)^3. - Franz Vrabec, Jun 24 2006
G.f.: ( 1+8*x ) / ( (16*x-1)*(4*x-1) ). - R. J. Mathar, Nov 29 2011
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MAPLE
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seq(sum((2*k-1)^3, k=1..2^n), n=0..15);
seq(sum(2^k, k=2*n..4*n), n=0..15);
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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