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A270835
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a(n) = smallest k such that A004431(n) +/- k are both positive squares.
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1
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4, 6, 12, 8, 16, 24, 10, 20, 30, 12, 24, 40, 36, 14, 48, 28, 42, 60, 16, 32, 48, 70, 64, 18, 36, 80, 54, 72, 96, 20, 40, 90, 60, 112, 80, 108, 22, 44, 66, 120, 88, 24, 110, 48, 140, 72, 132, 96, 160, 120, 26, 52, 78, 144, 180
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OFFSET
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1,1
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COMMENTS
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There can be more than one value k such that A004431(n) +/- k are both positive squares; i.e., when there are multiple ways to express A004431(n) as the sum of positive squares. These are the terms which appear more than once in A055096. For example A004431(19) = 65 = {(1^2 + 8^2), (4^2 + 7^2)}: 65 +/- 16 = {7^2, 9^2} and 65 +/- 56 = {3^2, 11^2}. So a(19) = 16 rather than 56.
Sequence contains every even number >=4 and no odd numbers.
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LINKS
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EXAMPLE
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a(11)=24 because A004431(11) = 40; 40+24 = 8^2 and 40-24 = 4^2.
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MATHEMATICA
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nn = 80; s = Select[Range[4 nn], Length[PowersRepresentations[#, 2,
2] /. {{0, _} -> Nothing, {a_, b_} /; a == b -> Nothing}] > 0 &]; Table[SelectFirst[Range[10 nn], And[IntegerQ@ Sqrt[s[[n]] + #], IntegerQ@ Sqrt[s[[n]] - #]] &], {n, nn}] (* Michael De Vlieger, Mar 24 2016, Version 10 *)
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PROG
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(PARI) issum(n)=if(n<5, return(0)); my(f=factor(n)%4); if(vecmin(f[, 1])>1, return(0)); for(i=1, #f[, 1], if(f[i, 1]==3 && f[i, 2]%2, return(0))); 1; \\ after A004431
findk(n) = {for (k=1, n, if (issquare(n+k) && issquare(n-k), return (k)); ); }
lista(nn) = {for (n=1, nn, if (issum(n), print1(findk(n), ", "); ); ); } \\ Michel Marcus, Mar 31 2016
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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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