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Search: a215799 -id:a215799
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Numbers k such that 2^k - 1 can be written in the form a^2 + 3*b^2.
+10
4
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
Amiram Eldar, Table of n, a(n) for n = 1..176 (terms 1..159 from V. Raman)
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
KEYWORD
nonn
AUTHOR
V. Raman, Aug 23 2012
EXTENSIONS
14 more terms from V. Raman, Aug 29 2012
STATUS
approved
Odd numbers k such that the Mersenne number 2^k - 1 can be written in the form a^2 + 3*b^2.
+10
4
3, 5, 7, 9, 13, 15, 17, 19, 21, 25, 27, 31, 37, 39, 45, 49, 51, 57, 61, 63, 65, 67, 75, 81, 85, 89, 93, 101, 103, 107, 111, 117, 125, 127, 133, 135, 139, 147, 153, 171, 183, 189, 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
OFFSET
1,1
COMMENTS
These 2^k - 1 numbers have no prime factors of the form 2 (mod 3) to an odd power.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..135 (terms 1..121 from V. Raman)
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&&i%2==1, print(i" -\t"a[1, ])))
CROSSREFS
KEYWORD
nonn
AUTHOR
V. Raman, Aug 23 2012
EXTENSIONS
8 more terms from V. Raman, Aug 28 2012
6 more terms from V. Raman, Aug 29 2012
STATUS
approved
Even numbers n such that 2^n - 1 can be written in the form a^2 + 3*b^2.
+10
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
OFFSET
1,1
COMMENTS
These 2^n-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
KEYWORD
nonn
AUTHOR
V. Raman, Aug 23 2012
EXTENSIONS
a(24)-a(47) from V. Raman, Aug 28 2012
STATUS
approved

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