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A224450
Numbers that are the primitive sum of two nonzero squares in exactly one way.
5
2, 5, 10, 13, 17, 25, 26, 29, 34, 37, 41, 50, 53, 58, 61, 73, 74, 82, 89, 97, 101, 106, 109, 113, 122, 125, 137, 146, 149, 157, 169, 173, 178, 181, 193, 194, 197, 202, 218, 226, 229, 233, 241, 250, 257, 269, 274, 277, 281, 289, 293, 298, 313, 314, 317, 337
OFFSET
1,1
COMMENTS
If one includes 1 as the first entry then this sequence gives the numbers that are the primitive sum of two squares (square 0 allowed) in exactly one way, if neither the order of the squares nor the signs of the numbers to be squared matters.
Compare this sequence with A025284.
If 2 is omitted from this sequence then all members are primitively represented by two distinct nonzero squares in exactly one way.
The sequence A193138(n), n >= 3, gives the multiplicities of the primitive sums of two squares (automatically distinct and nonzero for n >= 3 if such a sum exists at all).
Numbers such that there is exactly one pair (m,k) where m + k = a(n), and m*k == 1 (mod a(n)), m > 0 and m <= k. - Torlach Rush, Oct 19 2020
A pair (s,t) such that s+t = a(n) and s*t == +1 (mod a(n)) as above is obtained from a square root of -1 (mod a(n)) for s and t = a(n)-s. - Joerg Arndt, Oct 24 2020
FORMULA
This sequence gives the increasingly ordered numbers m which satisfy m = a^2 + b^2, with a and b integers, 0 < a <= b, gcd(a,b) = 1, and there is only one such representation, denoted by one doublet (a,b).
EXAMPLE
a(1) = 2 because m = 2 is the first number with a unique doublet (a,b) in question, namely (1,1) (gcd(1,1) = 1).
This is the only case with equal entries a and b (the non-distinct case).
8 is not a member of this sequence (but of A025284) because the only representation is 2^2 +2^2 and (2,2) is not primitive. Similarly for 18, 20, ...
a(2) = 5 because 5 is the second smallest number satisfying the given requirements. 3 and 4 have no representation as sum of two nonzero squares, and the unique doublet for 5 is (1,2) (with distinct a and b).
MATHEMATICA
nn = 20; t = Sort[Select[Flatten[Table[If[GCD[a, b] == 1, a^2 + b^2, 0], {a, nn}, {b, a, nn}]], 0 < # <= nn^2 &]]; t2 = Transpose[Select[Tally[t], #[[2]] == 1 &]][[1]] (* T. D. Noe, Apr 20 2013 *)
CROSSREFS
Cf. A025284, A008784 (primitive sums of two squares with square 0 included), A224770 (exactly 2 ways), A193138 (multiplicities).
Sequence in context: A103215 A037942 A008784 * A226828 A020893 A281292
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Apr 17 2013
STATUS
approved