A075127 Safe perfect powers: perfect powers n such that (n-1)/2 is also a perfect power.

9, 243, 289, 9801, 332929, 11309769, 384199201, 13051463049, 443365544449, 15061377048201, 511643454094369, 17380816062160329, 590436102659356801, 20057446674355970889, 681362750825443653409

Offset: 1

Author

Zak Seidov, Oct 11 2002

Keywords

• nonn

Comments

If both powers are squares, the smaller square is a triangular number, and all square triangular numbers (A001110) correspond to a member in this sequence. This proves that this sequence is infinite. Are there only finitely many other members, i.e., is A075127 \ A055792 finite? - Charles R Greathouse IV, Dec 12 2010

Crossrefs

Cf. A001110, A001597, A055792, A070428, A075114.

Programs

Mathematica

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

PARI

for(n=1, 1e10, if(ispower(n) && ispower((n-1)/2), print1(n, ", "))) \\ Altug Alkan, Oct 28 2015

Formula

Conjectures from Colin Barker, Oct 28 2015: (Start)

a(n) = 35*a(n-1)-35*a(n-2)+a(n-3) for n>5.

G.f.: x*(234*x^4-8182*x^3+7901*x^2+72*x-9) / ((x-1)*(x^2-34*x+1)).

(End)

Extensions

One more term from Robert G. Wilson v, Oct 16 2002

a(7)-a(15) from Donovan Johnson, Mar 10 2010