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A270835 a(n) = smallest k such that A004431(n) +/- k are both positive squares. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..55.

EXAMPLE

a(11)=24 because A004431(11) = 40; 40+24 = 8^2 and 40-24 = 4^2.

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A001844, A004431, A055096.

Sequence in context: A293836 A278252 A274298 * A272063 A152678 A110758

Adjacent sequences:  A270832 A270833 A270834 * A270836 A270837 A270838

KEYWORD

nonn

AUTHOR

Bob Selcoe, Mar 23 2016

STATUS

approved

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Last modified September 26 16:43 EDT 2020. Contains 337374 sequences. (Running on oeis4.)