A215798 Numbers k such that 2^k - 1 can be written in the form a^2 + 3*b^2.

2, 3, 5, 6, 7, 9, 13, 14, 15, 17, 18, 19, 21, 25, 26, 27, 31, 37, 38, 39, 42, 45, 49, 51, 54, 57, 61, 62, 63, 65, 67, 74, 75, 78, 81, 85, 89, 93, 98, 101, 103, 107, 111, 114, 117, 122, 125, 126, 127, 133, 134, 135, 139, 147, 153, 162, 171, 183, 186, 189, 195, 201, 217, 221, 222, 225, 234, 243, 254, 255, 257, 259, 267, 269, 271, 278, 279, 281, 293, 294

Offset

1,1

Comments

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

Links

V. Raman, Table of n, a(n) for n = 1..159

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, print(i" -\t"a[1, ])))

Crossrefs

Cf. A000043, A000225, A215799, A215806, A215807.

Sequence in context: A345982 A050747 A261514 * A074780 A292938 A056900

Adjacent sequences: A215795 A215796 A215797 * A215799 A215800 A215801

Keyword

nonn

Author

V. Raman, Aug 23 2012

Extensions

14 more terms from V. Raman, Aug 29 2012

Status

approved