This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A197594 #37 Jun 20 2024 16:58:01 %S A197594 3,7,15,127,511,1023,65535,2147483647,35184372088831, %T A197594 18014398509481983,18446744073709551615, %U A197594 3705346855594118253554271520278013051304639509300498049262642688253220148477951 %N A197594 Sum of the cubes of the first odd numbers up to a(n) equals the n-th perfect number. %C A197594 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. %D A197594 Albert H. Beiler: Recreations in the theory of numbers, New York, Dover, Second Edition, 1966, p. 22. %H A197594 Michel Marcus, <a href="/A197594/b197594.txt">Table of n, a(n) for n = 2..20</a> %F A197594 (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. %F A197594 a(n) = 2^((A000043(n)+1)/2) - 1. - _Charles R Greathouse IV_, Oct 17 2011 %F A197594 a(n) = sqrt(1 + sqrt(8*A000396(n) + 1)) - 1. - _Martin Renner_, Oct 17 2011 %F A197594 a(n) = 2^A138576(n) - 1. - _César Aguilera_, Apr 20 2024 %F A197594 a(n) = sqrt(2*(A000668(n)+1))-1 for n > 1. - _César Aguilera_, May 21 2024 %e A197594 a(2)=3, since 1^3 + 3^3 = 28, which is the second perfect number. %e A197594 a(3)=7, since 1^3 + 3^3 + 5^3 + 7^3 = 496, which is the third perfect number. %Y A197594 Cf. A000043, A000396, A065549, A138576. %K A197594 nonn %O A197594 2,1 %A A197594 _Martin Renner_, Oct 16 2011