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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. 9
4, 121, 144, 4900, 166464, 5654884, 192099600, 6525731524, 221682772224, 7530688524100, 255821727047184, 8690408031080164, 295218051329678400, 10028723337177985444, 340681375412721826704 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

REFERENCES

Mohammad K. Azarian, Diophantine Pair, Problem B-881, Fibonacci Quarterly, Vol. 37, No. 3, August 1999, pp. 277-278. Solution appeared in Vol. 38, No. 2, May 2000, pp. 183-184.

LINKS

Table of n, a(n) for n=1..15.

M. A. Bennett, Products of Consecutive Integers, Bull. London Math. Soc. 36 (2004), 683-694

FORMULA

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

MATHEMATICA

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

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 *)

CROSSREFS

Cf. A001597.

Cf. A117547 (square root of terms).

Sequence in context: A239187 A062081 A053881 * A017186 A098839 A227525

Adjacent sequences:  A075111 A075112 A075113 * A075115 A075116 A075117

KEYWORD

more,nonn

AUTHOR

Zak Seidov, Oct 11 2002

EXTENSIONS

Extended by Robert G. Wilson v, Oct 15 2002

More terms from T. D. Noe, Mar 29 2006

More terms from T. D. Noe, Nov 19 2006

STATUS

approved

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Last modified September 16 08:53 EDT 2019. Contains 327092 sequences. (Running on oeis4.)