A138576 -id:A138576 - cp's OEIS frontend

A108062 Numbers k such that 2^(2k-1) + 2^k + 1 is prime.

1, 2, 3, 6, 10, 15, 79, 82, 142, 190, 499, 5071, 7350, 38646, 42619, 53347

Offset: 1

Alexandre Wajnberg, Jun 02 2005

Next term >= 10^4. - Jeppe Stig Nielsen, Mar 28 2014

Jacques Basaldua, All Fermat and Mersenne numbers are squarefree.

Do[ If[ PrimeQ[2^(2n - 1) + 2^n + 1], Print[n]], {n, 5500}] (* Robert G. Wilson v, Jun 03 2005 *)

is(n)=ispseudoprime(2^(2*n-1)+2^n+1) \\ Charles R Greathouse IV, Jun 12 2017

a(n) = A006599(n) + 1.

a(1) and a(12) from Robert G. Wilson v, Jun 03 2005
a(13) from Jeppe Stig Nielsen, Mar 28 2014
a(14)-a(16) from Michael S. Branicky, Jan 02 2025

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

Original entry on oeis.org

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

Offset: 2

Martin Renner, Oct 16 2011

nonn

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.

			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.

Albert H. Beiler: Recreations in the theory of numbers, New York, Dover, Second Edition, 1966, p. 22.

Michel Marcus, Table of n, a(n) for n = 2..20

Cf. A000043, A000396, A065549, A138576.

(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

cp's OEIS Frontend

A108062 Numbers k such that 2^(2k-1) + 2^k + 1 is prime.

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