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%I #43 Jan 03 2022 20:37:34
%S 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,
%T 57,61,62,63,65,67,74,75,78,81,85,89,93,98,101,103,107,111,114,117,
%U 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
%N Numbers k such that 2^k - 1 can be written in the form a^2 + 3*b^2.
%C These 2^k - 1 numbers have no prime factors of the form 2 (mod 3) to an odd power.
%H V. Raman, <a href="/A215798/b215798.txt">Table of n, a(n) for n = 1..159</a>
%H Samuel S. Wagstaff, Jr., The Cunningham Project, <a href="http://homes.cerias.purdue.edu/~ssw/cun/">Factorizations of 2^n-1, for odd n's < 1200.</a>
%e 2^67 - 1 = 10106743618^2 + 3*3891344499^2 = 9845359982^2 + 3*4108642899^2.
%o (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,])))
%Y Cf. A000043, A000225, A215799, A215806, A215807.
%K nonn
%O 1,1
%A _V. Raman_, Aug 23 2012
%E 14 more terms from _V. Raman_, Aug 29 2012
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