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%I #37 Jan 04 2022 21:51:46
%S 3,5,7,9,13,15,17,19,21,25,27,31,37,39,45,49,51,57,61,63,65,67,75,81,
%T 85,89,93,101,103,107,111,117,125,127,133,135,139,147,153,171,183,189,
%U 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
%N Odd numbers k such that the Mersenne number 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="/A215806/b215806.txt">Table of n, a(n) for n = 1..121</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&&i%2==1, print(i" -\t"a[1, ])))
%Y Cf. A000043, A000225, A215798, A215799, A215807.
%K nonn
%O 1,1
%A _V. Raman_, Aug 23 2012
%E 8 more terms from _V. Raman_, Aug 28 2012
%E 6 more terms from _V. Raman_, Aug 29 2012
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