A094740 -id:A094740 - cp's OEIS frontend

A025321 Numbers that are the sum of 3 nonzero squares in exactly 1 way.

3, 6, 9, 11, 12, 14, 17, 18, 19, 21, 22, 24, 26, 29, 30, 34, 35, 36, 42, 43, 44, 45, 46, 48, 49, 50, 53, 56, 61, 65, 67, 68, 70, 72, 73, 76, 78, 82, 84, 88, 91, 93, 96, 97, 104, 106, 109, 115, 116, 120, 133, 136, 140, 142, 144, 145, 157, 163, 168, 169, 172, 176, 180, 184, 190

Offset: 1

David W. Wilson

nonn

It appears that all terms have the form 4^i A094740(j) for some i and j. - T. D. Noe, Jun 06 2008

This is true, because A025427(4*n) = A025427(n) for all n. - Robert Israel, Mar 09 2016

Donovan Johnson, Table of n, a(n) for n = 1..605 (terms < 10^8; first 417 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to sums of squares

Cf. A000408, A025427, A243148.

lim=20; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && nT. D. Noe, Jun 06 2008 *)
b[n_, i_, k_, t_] := b[n, i, k, t] = If[n == 0, If[t == 0, 1, 0], If[i<1 || t<1, 0, b[n, i - 1, k, t] + If[i^2 > n, 0, b[n - i^2, i, k, t - 1]]]];
T[n_, k_] := b[n, Sqrt[n] // Floor, k, k];
Position[Table[T[n, 3], {n, 0, 200}], 1] - 1 // Flatten (* Jean-François Alcover, Nov 06 2020, after Alois P. Heinz in A243148 *)

is(n)=if(n<11, return(n>0 && n%3==0)); if(n%4==0, return(is(n/4))); my(w); for(i=sqrtint((n-1)\3)+1,sqrtint(n-2), my(t=n-i^2); for(j=sqrtint((t-1)\2)+1,min(sqrtint(t-1),i), if(issquare(t-j^2), w++>1 && return(0)))); w \\ Charles R Greathouse IV, Aug 05 2024

A243148(a(n),3) = 1. - Alois P. Heinz, Feb 25 2019

A025414 a(n) is the smallest number that is the sum of 3 nonzero squares in exactly n ways.

Original entry on oeis.org

3, 27, 54, 129, 194, 209, 341, 374, 614, 594, 854, 1106, 1314, 1154, 1286, 1746, 1634, 1881, 2141, 2246, 2609, 2889, 3461, 3366, 3449, 3506, 4241, 4289, 5066, 4826, 5381, 5606, 6569, 5561, 6254, 7601, 8186, 8069, 8714, 8126, 9434, 8921, 8774, 11066, 11574

Offset: 1

David W. Wilson

nonn

A025427(a(n)) = n and A025427(m) != n for m < a(n). - Reinhard Zumkeller, Feb 26 2015

			54 is the smallest number having three partitions into nonzero squares: 54 = 1+4+49 = 4+25+25 = 9+9+36.

Donovan Johnson, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of squares

Cf. A094740 (n having a unique partition into three positive squares), A095812 (greatest number having exactly n partitions into three positive squares).

Cf. A025427.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a025414 = fromJust . (`elemIndex` a025427_list)
-- Reinhard Zumkeller, Feb 26 2015

lim=200; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && n

A094739 Numbers m such that 4^k*m, for integer k >= 0, are numbers having a unique partition into three squares.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 14, 19, 21, 22, 30, 35, 37, 42, 43, 46, 58, 67, 70, 78, 91, 93, 115, 133, 142, 163, 190, 235, 253, 403, 427

Offset: 1

T. D. Noe, May 24 2004

Lehmer's paper has an erroneous version of this sequence. He omits 163 and includes 162 (which has 4 partitions) and 182 (which has 3 partitions). Lehmer conjectures that there are no more terms. Note that squares are allowed to be zero.
From Wolfdieter Lang, Aug 27 2020: (Start)
Another name is: Integers not divisible by 4 that are uniquely represented as x^2 + y^2 + z^2 with integers 0 <= x <= y <= z.
This sequence of 33 numbers is complete. See Arno, Theorem 8, p. 332, where 19 is missing, as observed by Kaplansky, Remark 2.1. (a) - (c), p. 87.
All positive integers represented uniquely as sum of three squares of nonnegative numbers, ignoring order and signs, are given by 4^k*a(n), for integer k >= 0 and n = 1 .. 33. See Arno, also p. 322, with some known results, and Kaplansky's Remark 2.1.(c). (End)

			The unique partitions of m*4^k into three squares are,
for m = 1:
1 = 1^2 + 0^2 + 0^2;
4 = 2^2 + 0^2 + 0^2;
16 = 4^2 + 0^2 + 0^2;
...
for m = 163:
163 = 9^2 + 9^2 + 1^2;
163*4 = 18^2 + 18^2 + 2^2;
163*16 = 36^2 + 36^2 + 4^2;
...

Steven Arno, The imaginary quadratic fields of class number 4, Acta Arithmetica 60.4 (1992) 321 - 334.
Irving Kaplansky, Integers Uniquely Represented by Certain Ternary Forms, in "The Mathematics of Paul Erdős I", Ronald. L. Graham and Jaroslav Nešetřil (Eds.), Springer, 1997, pp. 86 - 94.
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No.8, October 1948, pp. 476-481.

Cf. A005875 (number of ways of writing n as the sum of three squares), A094740 (n having a unique partition into three positive squares).

lim=100; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && n0&]

Keyword full added by Wolfdieter Lang, Aug 27 2020

A096017 Numbers n such that 4^k*n, for k >= 0, have a unique partition into three distinct positive squares.

Original entry on oeis.org

14, 21, 26, 29, 30, 35, 38, 41, 42, 45, 46, 49, 50, 53, 54, 59, 61, 65, 66, 70, 75, 78, 81, 83, 91, 93, 106, 107, 109, 113, 114, 115, 118, 121, 133, 137, 139, 142, 145, 147, 153, 157, 162, 169, 171, 178, 190, 198, 202, 205, 211, 214, 219, 226, 235, 243, 253, 258, 262, 265, 277, 283, 289, 291, 298, 307, 313, 323, 331, 337, 358, 363, 379, 387, 397, 403, 418, 427, 438, 442, 445, 457, 466, 498, 499, 505, 547, 562, 577, 603, 643, 723, 793, 883, 907, 1003, 1227, 1243, 1387, 1411, 1467, 1507

Offset: 1

T. D. Noe, Jun 15 2004

It is conjectured that this sequence is complete.

			793 is in this sequence because 793 = 6^2 + 9^2 + 26^2 is the unique partition of 793.

Cf. A094739 (primitive n having a unique partition into three squares), A094740 (primitive n having a unique partition into three positive squares).

lim=100; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && n0&]

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A025321 Numbers that are the sum of 3 nonzero squares in exactly 1 way.

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A025414 a(n) is the smallest number that is the sum of 3 nonzero squares in exactly n ways.

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A094739 Numbers m such that 4^k*m, for integer k >= 0, are numbers having a unique partition into three squares.

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A096017 Numbers n such that 4^k*n, for k >= 0, have a unique partition into three distinct positive squares.

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