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A270820
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Prime powers p^k (p prime, k > 1) that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.
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0
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343, 12167, 16807, 29791, 103823, 357911, 493039, 823543, 1092727, 2048383, 3442951, 4657463, 6436343, 6967871, 7880599, 11089567, 13651919, 18191447, 19902511, 28629151, 30080231, 40353607, 46268279, 49430863, 56181887, 80062991, 84604519, 99252847
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OFFSET
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1,1
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COMMENTS
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Proper prime powers that are the sum of 4 but no fewer nonzero squares.
This sequence lists the numbers of the form A007522(n)^(2*k+1) where n,k > 0.
Subsequence of A267321.
Terms are 7^3, 23^3, 7^5, 31^3, 47^3, 71^3, 79^3, 7^7, 103^3, 127^3, 151^3, 167^3, 23^5, 191^3, 199^3, ...
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LINKS
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Table of n, a(n) for n=1..28.
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EXAMPLE
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16807 is a term because 16807 = 7^5 and there is no integer values of x, y and z for the equation 7^5 = x^2 + y^2 + z^2.
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MATHEMATICA
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nn = 120; Select[TakeWhile[Union@ Flatten@ Map[Prime[Range@ nn]^# &, Range[2, Floor[Log2[PrimePi@ nn]^2]]], # <= Prime[nn]^2 &], ! Resolve[Exists[{x, y, z}, Reduce[# == x^2 + y^2 + z^2, {x, y, z}, Integers]]] &] (* Michael De Vlieger, Mar 23 2016 *)
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PROG
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(PARI) isA004215(n) = {my(fouri, j) ; fouri=1 ; while(n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if(j % 8==0, return(1)) ; ) ; fouri *= 4 ; ) ; return(0) ; }
forcomposite(n=4, 1e7, if(isA004215(n) && isprimepower(n), print1(n, ", ")));
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CROSSREFS
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Cf. A000961, A004215, A007522, A267321, A246547.
Sequence in context: A277636 A224427 A134263 * A140250 A117197 A269554
Adjacent sequences: A270817 A270818 A270819 * A270821 A270822 A270823
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KEYWORD
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nonn
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AUTHOR
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Altug Alkan, Mar 23 2016
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STATUS
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approved
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