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

2,1

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

Table of n, a(n) for n=2..13.

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

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,changed

Author

Martin Renner, Oct 16 2011

Status

approved