This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A075114 #26 Feb 16 2024 06:33:46
%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 Catalan'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_, Mar 29 2006
%H A075114 M. A. Bennett, <a href="https://www.math.ubc.ca/~bennett/B-BLMS.pdf">Products of Consecutive Integers</a>, Bull. London Math. Soc. 36 (2004), 683-694
%F A075114 Empirical G.f.: x*(117*x^4-4091*x^3+3951*x^2+19*x-4) / ((x-1)*(x^2-34*x+1)). - _Colin Barker_, Dec 21 2012
%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_, Mar 29 2006 *)
%Y A075114 Cf. A001597.
%Y A075114 Cf. A117547 (square root of terms).
%K A075114 more,nonn
%O A075114 1,1
%A A075114 _Zak Seidov_, Oct 11 2002
%E A075114 Extended by _Robert G. Wilson v_, Oct 15 2002
%E A075114 More terms from _T. D. Noe_, Mar 29 2006
%E A075114 More terms from _T. D. Noe_, Nov 19 2006