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
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
KEYWORD
nonn,more
AUTHOR
V. Raman, Aug 23 2012
EXTENSIONS
a(23)-a(47) from Robert G. Wilson v, Aug 26 2012
a(48) from Amiram Eldar, Oct 18 2024
STATUS
approved