A215799 Prime numbers p such that the Mersenne number 2^p - 1 can be written in the form a^2 + 3*b^2.

2, 3, 5, 7, 13, 17, 19, 31, 37, 61, 67, 89, 101, 103, 107, 127, 139, 257, 269, 271, 281, 293, 347, 349, 353, 373, 379, 401, 457, 461, 499, 521, 523, 569, 577, 607, 631, 647, 727, 751, 863, 881, 907, 983, 1039, 1061, 1063

Offset

1,1

Comments

These numbers have no prime factors congruent to 2 (mod 3) raised to an odd power. Prime factors which are == 2 (mod 3) come in pairs.

Mersenne exponents, A000043, are a proper subset.

There are no other terms <= 1116. - Robert G. Wilson v, Aug 26 2012

Links

Table of n, a(n) for n=1..47.

Will Edgington's Mersenne Page, Mersenne and Prime numbers and Quick links to other related sites.

Paul C. Leyland, Factorizations of 2^n-1, for odd n's < 1200.

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.

Mathematica

fQ[n_] := Union[ Mod[ Transpose[ FactorInteger[2^n - 1]][[1]], 3]] == {1}; p = 2; lst = {}; While[p < 300, If[ fQ@ p, AppendTo[lst, p]; Print@ p]; p = NextPrime@ p] (* Or *)
p=2; (* open the first or second link and copy the listed factors for the prime exponent and paste into the parentheses that follow and change any periods to commas *) p = NextPrime@ p; pf = {}; Mod[Flatten[{pf, (2^p - 1)/Times @@ pf}], 3] (* Robert G. Wilson v, Aug 26 2012 *)

Prog

(PARI) forprime(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, A215798, A215806, A215807.

Sequence in context: A293008 A038903 A136003 * A237057 A183073 A123856

Adjacent sequences: A215796 A215797 A215798 * A215800 A215801 A215802

Keyword

nonn

Author

V. Raman, Aug 23 2012

Extensions

a(23)-a(47) from Robert G. Wilson v, Aug 26 2012

Status

approved