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A197594 Sum of the cubes of the first odd numbers up to a(n) equals the n-th perfect number. 0
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

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]

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.

Sequence in context: A193831 A246719 A077775 * A206851 A033089 A175878

Adjacent sequences:  A197591 A197592 A197593 * A197595 A197596 A197597

KEYWORD

nonn

AUTHOR

Martin Renner, Oct 16 2011

STATUS

approved

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Last modified December 7 23:55 EST 2016. Contains 278902 sequences.