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A075127
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Safe perfect powers: perfect powers n such that (n-1)/2 is also a perfect power.
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1
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9, 243, 289, 9801, 332929, 11309769, 384199201, 13051463049, 443365544449, 15061377048201, 511643454094369, 17380816062160329, 590436102659356801, 20057446674355970889, 681362750825443653409
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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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)
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MATHEMATICA
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pp = Select[ Range[10^8], Apply[ GCD, Last[ Transpose[ FactorInteger[ # ]]]] > 1 & ]; Select[pp, Apply[GCD, Last[ Transpose[ FactorInteger[( # - 1)/2]]]] > 1 & ]
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PROG
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(PARI) for(n=1, 1e10, if(ispower(n) && ispower((n-1)/2), print1(n, ", "))) \\ Altug Alkan, Oct 28 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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