A224450 Numbers that are the primitive sum of two nonzero squares in exactly one way.

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

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

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

Adjacent sequences: A224447 A224448 A224449 * A224451 A224452 A224453

Keyword

nonn

Author

Wolfdieter Lang, Apr 17 2013

Status

approved