%I A075114
%S A075114 4,121,144,4900,166464,5654884,192099600,6525731524,221682772224,
%T A075114 7530688524100,255821727047184,8690408031080164,295218051329678400,
%U A075114 10028723337177985444,340681375412721826704
%N A075114 Perfect powers n such that 2n+1 is a perfect power; the value of y^b 
               in the solution of the Diophantine equation x^a-2y^b=1.
%C A075114 Note that the first ten numbers in this sequence are all squares. Except 
               for 121, these squares are the y^2 in the Pell equation x^2-2y^2=1, 
               whose solutions (x,y) are in sequences A001541 and A001542. The equation 
               x^a-2y^b=1 is very similar to Calalan's equation x^a-y^b=1, which 
               has only one solution. Bennett shows that the equation x^2-2y^b=1 
               has no solutions for b>2. Hence all the terms in this sequence are 
               squares and solutions other than the Pell solutions must satisfy 
               x^a-2y^2=1 for a>2. The one known solution is 3^5-2*11^2=1. Are there 
               any others? - T. D. Noe (noe(AT)sspectra.com), Mar 29 2006
%D A075114 M. A. Bennett, Products of Consecutive Integers, Bull. London Math. Soc. 
               36 (2004), 683-694.
%t A075114 pp = Select[ Range[10^8], Apply[ GCD, Last[ Transpose[ FactorInteger[ 
               # ]]]] > 1 & ]; Select[pp, Apply[GCD, Last[ Transpose[ FactorInteger[2# 
               + 1]]]] > 1 & ]
%t A075114 lim=10^14; lst={}; k=2; While[n=Floor[lim^(1/k)]; n>1, lst=Join[lst,Range[2,
               n]^k]; k++ ]; lst=Union[lst]; Intersection[lst,(lst-1)/2] - T. D. 
               Noe (noe(AT)sspectra.com), Mar 29 2006
%Y A075114 Cf. A001597.
%Y A075114 Cf. A117547 (square root of terms).
%Y A075114 Sequence in context: A006607 A062081 A053881 this_sequence A017186 A098839 
               A071129
%Y A075114 Adjacent sequences: A075111 A075112 A075113 this_sequence A075115 A075116 
               A075117
%K A075114 more,nonn
%O A075114 1,1
%A A075114 Zak Seidov (zakseidov(AT)yahoo.com), Oct 11 2002
%E A075114 Extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 15 2002
%E A075114 More terms from T. D. Noe (noe(AT)sspectra.com), Mar 29 2006
%E A075114 More terms from T. D. Noe (noe(AT)sspectra.com), Nov 19 2006