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

5, 10, 13, 17, 20, 26, 29, 34, 37, 40, 41, 45, 52, 53, 58, 61, 68, 73, 74, 80, 82, 89, 90, 97, 101, 104, 106, 109, 113, 116, 117, 122, 136, 137, 146, 148, 149, 153, 157, 160, 164, 173, 178, 180, 181, 193, 194, 197, 202, 208, 212, 218, 226, 229, 232, 233, 234, 241, 244, 245, 257, 261, 269, 272

Offset: 1

Author

Jean-Christophe Hervé, Nov 16 2013

Keywords

• nonn

Comments

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.

These are the primitive elements of A004431, the integers which are the sum of two nonzero distinct squares.

Numbers such that A004018(a(n)) = 8.

The square of these numbers is also uniquely decomposable into a sum of two squares, thus this sequence is a subsequence of A084645.

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.

Numbers n such that n^3 is the sum of two nonzero squares in exactly two ways. - Altug Alkan, Jul 01 2016

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

Examples

			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).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000 (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)
• Eric Weisstein's World of Mathematics, Square Number
• Wikipedia (fr), Théorème des deux carrés
• G. Xiao, Two squares
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A001481, A004431, A002144, A028982, A353813 (characteristic function).

Subsequence of A004431, of A084645, of A125022, and of A191217.

Cf. A004018, A022554 (A004018 = 0), A125853 (A004018 = 4).

Programs

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

Formula

Terms are obtained by the products A125853(k)*A002144(p) for k, p > 0, ordered by increasing values.

{k | A004018(k) = 8}.