1,1

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

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

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 (noe(AT)sspectra.com), Mar 29 2006

Cf. A001597.

Cf. A117547 (square root of terms).

Sequence in context: A006607 A062081 A053881 this_sequence A017186 A098839 A071129

Adjacent sequences: A075111 A075112 A075113 this_sequence A075115 A075116 A075117

more,nonn

Zak Seidov (zakseidov(AT)yahoo.com), Oct 11 2002

Extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 15 2002

More terms from T. D. Noe (noe(AT)sspectra.com), Mar 29 2006

More terms from T. D. Noe (noe(AT)sspectra.com), Nov 19 2006