A215806 Odd numbers k such that the Mersenne number 2^k - 1 can be written in the form a^2 + 3*b^2.

3, 5, 7, 9, 13, 15, 17, 19, 21, 25, 27, 31, 37, 39, 45, 49, 51, 57, 61, 63, 65, 67, 75, 81, 85, 89, 93, 101, 103, 107, 111, 117, 125, 127, 133, 135, 139, 147, 153, 171, 183, 189, 195, 201, 217, 221, 225, 243, 255, 257, 259, 267, 269, 271, 279, 281, 293, 303, 309, 321, 333, 343, 347, 349, 351, 353, 373, 375, 379, 381, 399

Offset

1,1

Comments

These 2^k - 1 numbers have no prime factors of the form 2 (mod 3) to an odd power.

Links

Amiram Eldar, Table of n, a(n) for n = 1..135 (terms 1..121 from V. Raman)

Samuel S. Wagstaff, Jr., The Cunningham Project, Factorizations of 2^n-1, for odd n's < 1200.

Example

2^67 - 1 = 10106743618^2 + 3*3891344499^2 = 9845359982^2 + 3*4108642899^2.

Prog

(PARI) for(i=2, 100, a=factorint(2^i-1)~; has=0; for(j=1, #a, if(a[1, j]%3==2&&a[2, j]%2==1, has=1; break)); if(has==0&&i%2==1, print(i" -\t"a[1, ])))

Crossrefs

Cf. A000043, A000225, A215798, A215799, A215807.

Sequence in context: A204454 A029740 A063425 * A333369 A064608 A024893

Adjacent sequences: A215803 A215804 A215805 * A215807 A215808 A215809

Keyword

nonn

Author

V. Raman, Aug 23 2012

Extensions

8 more terms from V. Raman, Aug 28 2012

6 more terms from V. Raman, Aug 29 2012

Status

approved