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A271536 Integers n such that n^2-1 and n^2 are the sum of two nonzero squares. 0
17, 35, 51, 73, 105, 145, 195, 233, 273, 289, 291, 339, 451, 465, 521, 577, 579, 585, 611, 675, 723, 777, 801, 809, 819, 899, 915, 969, 1043, 1059, 1097, 1153, 1155, 1185, 1225, 1281, 1313, 1347, 1353, 1395, 1545, 1601, 1603, 1665, 1683, 1731, 1745, 1763, 1801, 1873, 1923, 1961, 1971, 2019 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Corresponding n^2 values are 289, 1225, 2601, 5329, 11025, 21025, 38025, 54289, 74529, 83521, 84681, ...
If n is in this sequence, so is n^2. Proof:
n is a term of this sequence if and only if n^2 = a^2 + b^2 and n^2-1 = c^2 + d^2 for a,b,c,d are nonzero integers.
If n^2 = a^2 + b^2, then n^4 = (n^2)*(n^2) = (n^2)*(a^2 + b^2) = (n*a)^2 + (n*b)^2.
If n^2-1 = c^2 + d^2, then n^4-1 = (n^2-1)*(n^2+1) = (c^2 + d^2)*(n^2+1) = (c*n + d)^2 + (d*n - c)^2. Note that (d*n - c) > 0 because of definition of n.
Since both (n^2)^2 and (n^2)^2-1 are the sum of two nonzero squares, n^2 must be a term.
With repeating of same procedure, it can be seen that if n is in this sequence, so is n^(2^k), for k >= 0.
LINKS
EXAMPLE
17 is a term because 17^2 = 8^2 + 15^2 and 17^2 - 1 = 12^2 + 12^2.
MATHEMATICA
fQ[n_] := Length[PowersRepresentations[n, 2, 2] /. {0, _} -> Nothing] > 0; Select[Range@ 2020, And[fQ[#^2 - 1], fQ[#^2]] &] (* Michael De Vlieger, Apr 10 2016 *)
PROG
(PARI) isA000404(n) = {for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
for(n=1, 1e4, if(isA000404(n^2-1) && isA000404(n^2), print1(n, ", ")))
CROSSREFS
Sequence in context: A110287 A102813 A360573 * A041568 A042305 A041566
KEYWORD
nonn
AUTHOR
Altug Alkan, Apr 09 2016
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)