A197594 Sum of the cubes of the first odd numbers up to a(n) equals the n-th perfect number.

3, 7, 15, 127, 511, 1023, 65535, 2147483647, 35184372088831, 18014398509481983, 18446744073709551615, 3705346855594118253554271520278013051304639509300498049262642688253220148477951

Offset: 2

Author

Martin Renner, Oct 16 2011

Keywords

• nonn

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.

Examples

			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.
		

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

Crossrefs

Cf. A000043, A000396, A065549, A138576.

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