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 A197594 Sum of the cubes of the first odd numbers up to a(n) equals the n-th perfect number. 1
 3, 7, 15, 127, 511, 1023, 65535, 2147483647, 35184372088831, 18014398509481983, 18446744073709551615, 3705346855594118253554271520278013051304639509300498049262642688253220148477951 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS Except for the first perfect number 6, every even perfect number 2^(p-1)*(2^p - 1) is the sum of the cubes of the first 2^((p-1)/2) odd numbers. REFERENCES Albert H. Beiler: Recreations in the theory of numbers, New York, Dover, Second Edition, 1966, p. 22. LINKS Michel Marcus, Table of n, a(n) for n = 2..20 FORMULA (1/8)*(a(n) + 1)^2*(a(n)^2 + 2*a(n) - 1) = 2^(p-1)*(2^p - 1) with p = 2*log(a(n) + 1)/log(2) - 1 a Mersenne prime. a(n) = 2^((A000043(n)+1)/2) - 1. - Charles R Greathouse IV, Oct 17 2011 a(n) = sqrt(1 + sqrt(8*A000396(n) + 1)) - 1. - Martin Renner, Oct 17 2011 a(n) = 2^A138576(n) - 1. - César Aguilera, Apr 20 2024 a(n) = sqrt(2*(A000668(n)+1))-1 for n > 1. - César Aguilera, May 21 2024 EXAMPLE a(2)=3, since 1^3 + 3^3 = 28, which is the second perfect number. a(3)=7, since 1^3 + 3^3 + 5^3 + 7^3 = 496, which is the third perfect number. CROSSREFS Cf. A000043, A000396, A065549, A138576. Sequence in context: A193831 A246719 A077775 * A206851 A033089 A370661 Adjacent sequences: A197591 A197592 A197593 * A197595 A197596 A197597 KEYWORD nonn AUTHOR Martin Renner, Oct 16 2011 STATUS approved

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Last modified July 20 18:59 EDT 2024. Contains 374459 sequences. (Running on oeis4.)