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A274356
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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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0
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1, 5, 16, 25, 41, 80, 81, 125, 256, 400, 405, 425, 625, 656, 841, 1225, 1280, 1296, 1681, 2000, 2025, 2401, 3125, 3321, 3721, 4096, 6400, 6480, 6561, 6800, 8281, 8381, 10000, 10125, 10496, 12005, 13456, 14161, 14641, 15625, 19600, 20480, 20736, 25625
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OFFSET
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1,2
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COMMENTS
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Numbers n such that n^k is the average of a nonzero square and a nonzero fourth power for all k > 0.
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.
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.
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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LINKS
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EXAMPLE
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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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PROG
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(PARI) isA111925(n)=for(b=1, sqrtnint(n-1, 4), if(issquare(n-b^4), return(1))); 0;
lista(nn)=for(n=1, nn, if(isA111925(2*n) && isA111925(2*n^2) && isA111925(2*n^3), print1(n, ", ")));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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