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Numbers n such that n^k is of the form (a^2 + b^4)/2 for all k > 0 (a, b > 0).

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`%I #11 Jun 21 2016 16:20:30
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`%S 1,5,16,25,41,80,81,125,256,400,405,425,625,656,841,1225,1280,1296,
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`%T 1681,2000,2025,2401,3125,3321,3721,4096,6400,6480,6561,6800,8281,
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`%U 8381,10000,10125,10496,12005,13456,14161,14641,15625,19600,20480,20736,25625
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`%N Numbers n such that n^k is of the form (a^2 + b^4)/2 for all k > 0 (a, b > 0).
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`%C Numbers n such that n^k is the average of a nonzero square and a nonzero fourth power for all k > 0.
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`%C If n^k = (a^2 + b^4)/2, then n^(k+4) = ((n^2*a)^2 + (n*b)^4)/2. So this sequence lists numbers n such that 2*n, 2*n^2 and 2*n^3 are in A111925.
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`%C If n is in this sequence, then n^t is also in this sequence for all t > 1. So in this sequence there are infinitely many prime powers.
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`%C There are infinitely many (x, y) pairs in this sequence where x and y are distinct terms of this sequence such that x*y is also in this sequence.
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`%e 5 is a term because 5 = (3^2 + 1^4)/2, 5^2 = (7^2 + 1^4)/2, 5^3 = (13^2 + 3^4)/2, 5^4 = ((5^2)^2 + 5^4)/2.
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`%o (PARI) isA111925(n)=for(b=1, sqrtnint(n-1, 4), if(issquare(n-b^4), return(1))); 0;
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`%o lista(nn)=for(n=1, nn, if(isA111925(2*n) && isA111925(2*n^2) && isA111925(2*n^3), print1(n, ", ")));
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`%Y Cf. A111925.
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`%K nonn
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`%O 1,2
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`%A _Altug Alkan_, Jun 18 2016
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