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URL: https://oeis.org/A215807

⇱ A215807 - OEIS

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A215807
Even numbers k such that 2^k - 1 can be written in the form a^2 + 3*b^2.
3
2, 6, 14, 18, 26, 38, 42, 54, 62, 74, 78, 98, 114, 122, 126, 134, 162, 186, 222, 234, 254, 278, 294, 342, 366, 378, 402, 434, 486, 518, 558, 666, 702, 762, 834, 882, 914, 1026, 1098, 1134, 1206, 1302, 1458, 1554, 1674, 1998, 2106, 2286, 2502, 2646
OFFSET
1,1
COMMENTS
These 2^k-1 numbers have no prime factors of the form 2 (mod 3) to an odd power.
LINKS
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
Select[Range[2, 200, 2], Length[FindInstance[x^2 + 3*y^2 == 2^# - 1, {x, y}, Integers]] > 0 &] (* G. C. Greubel, Apr 14 2017 *)
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==0, print(i" -\t"a[1, ])))
CROSSREFS
Even terms of A215798.
Sequence in context: A139269 A186299 A163777 * A373437 A140525 A189804
KEYWORD
nonn
AUTHOR
V. Raman, Aug 23 2012
EXTENSIONS
a(24)-a(47) from V. Raman, Aug 28 2012
a(48)-a(50) from Max Alekseyev, Oct 23 2025
STATUS
approved