A224450 - OEIS

%I #38 Dec 30 2024 17:27:54

%S 2,5,10,13,17,25,26,29,34,37,41,50,53,58,61,73,74,82,89,97,101,106,

%T 109,113,122,125,137,146,149,157,169,173,178,181,193,194,197,202,218,

%U 226,229,233,241,250,257,269,274,277,281,289,293,298,313,314,317,337

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

%C 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.

%C Compare this sequence with A025284.

%C If 2 is omitted from this sequence then all members are primitively represented by two distinct nonzero squares in exactly one way.

%C 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).

%C 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

%C 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

%H T. D. Noe, <a href="/A224450/b224450.txt">Table of n, a(n) for n = 1..10000</a>

%F 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).

%e 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).

%e This is the only case with equal entries a and b (the non-distinct case).

%e 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, ...

%e 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).

%t 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 *)

%Y Cf. A025284, A008784 (primitive sums of two squares with square 0 included), A224770 (exactly 2 ways), A193138 (multiplicities).

%K nonn

%O 1,1

%A _Wolfdieter Lang_, Apr 17 2013