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A230779 Numbers which are uniquely decomposable into a sum of two squares, the unique decomposition being with two distinct nonzero squares.

%I A230779 #71 Feb 16 2025 08:33:20
%S A230779 5,10,13,17,20,26,29,34,37,40,41,45,52,53,58,61,68,73,74,80,82,89,90,
%T A230779 97,101,104,106,109,113,116,117,122,136,137,146,148,149,153,157,160,
%U A230779 164,173,178,180,181,193,194,197,202,208,212,218,226,229,232,233,234,241,244,245,257,261,269,272
%N A230779 Numbers which are uniquely decomposable into a sum of two squares, the unique decomposition being with two distinct nonzero squares.
%C A230779 Numbers with exactly one prime factor of form 4*k+1, that must have multiplicity one, and no prime factor of the form 4*k+3 with odd multiplicity. There is thus no square in the sequence.
%C A230779 These are the primitive elements of A004431, the integers which are the sum of two nonzero distinct squares.
%C A230779 Numbers such that A004018(a(n)) = 8.
%C A230779 The square of these numbers is also uniquely decomposable into a sum of two squares, thus this sequence is a subsequence of A084645.
%C A230779 Also a subsequence of A191217: the two sequences are equal up to a(76) = 320, then A191217(77) = 325, the value which is missing from this sequence, as a(77) = 328 = A191217(78). (3125 is also missing from this sequence, although present in A191217, and it is the 31st such number). - Corrected by _Antti Karttunen_, May 14 2022.
%C A230779 Numbers n such that n^3 is the sum of two nonzero squares in exactly two ways. - _Altug Alkan_, Jul 01 2016
%C A230779 Sequence A125022 (numbers with a unique partition as the sum of 2 squares x^2 + y^2), but without any terms of A028982 (squares and twice squares) that might occur there. - _Antti Karttunen_, May 14 2022
%H A230779 Antti Karttunen, <a href="/A230779/b230779.txt">Table of n, a(n) for n = 1..20000</a> (extending the previous b-file from Jean-Christophe Hervé, which contained terms up to the 1647th term 10009, but accidentally missed terms 8992 and 9376)
%H A230779 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareNumber.html">Square Number</a>
%H A230779 Wikipedia (fr), <a href="http://fr.wikipedia.org/wiki/Théorème_des_deux_carrés_de_Fermat">Théorème des deux carrés</a>
%H A230779 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~twosquares.en.html">Two squares</a>
%H A230779 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A230779 Terms are obtained by the products A125853(k)*A002144(p) for k, p > 0, ordered by increasing values.
%F A230779 {k | A004018(k) = 8}.
%e A230779 a(1) = 5 = 4+1, a(2) = 10 = 9+1, a(3) =  13 = 9+4. However 2 = 1+1, 4 = 4+0, 8 = 4+4 are excluded because the unique decomposition of these numbers in two squares is not with two distinct nonzero squares; 25, 50, 100 are also excluded because there are two decompositions of these numbers in two squares (including one with equal or zero squares).
%o A230779 (PARI) isok(n) = {f = factor(n); nb1 = 0; for (i=1, #f~, p = f[i, 1]; ep = f[i, 2]; if (p % 4 == 1, nb1 ++; if (ep != 1, return (0))); if (p % 4 == 3, if (ep % 2, return (0)));); return (nb1 == 1);} \\ _Michel Marcus_, Nov 17 2013
%Y A230779 Cf. A001481, A004431, A002144, A028982, A353813 (characteristic function).
%Y A230779 Subsequence of A004431, of A084645, of A125022, and of A191217.
%Y A230779 Cf. A004018, A022554 (A004018 = 0), A125853 (A004018 = 4).
%K A230779 nonn
%O A230779 1,1
%A A230779 _Jean-Christophe Hervé_, Nov 16 2013