A075114 - OEIS

%I #26 Feb 16 2024 06:33:46

%S 4,121,144,4900,166464,5654884,192099600,6525731524,221682772224,

%T 7530688524100,255821727047184,8690408031080164,295218051329678400,

%U 10028723337177985444,340681375412721826704

%N 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 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 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 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 pp = Select[ Range[10^8], Apply[ GCD, Last[ Transpose[ FactorInteger[ # ]]]] > 1 & ]; Select[pp, Apply[GCD, Last[ Transpose[ FactorInteger[2# + 1]]]] > 1 & ]

%t 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 Cf. A001597.

%Y Cf. A117547 (square root of terms).

%K more,nonn

%O 1,1

%A _Zak Seidov_, Oct 11 2002

%E Extended by _Robert G. Wilson v_, Oct 15 2002

%E More terms from _T. D. Noe_, Mar 29 2006

%E More terms from _T. D. Noe_, Nov 19 2006