A025415 -id:A025415 - cp's OEIS frontend

A024803 Numbers that are the sum of 3 distinct nonzero squares, including repetitions.

14, 21, 26, 29, 30, 35, 38, 41, 42, 45, 46, 49, 50, 53, 54, 56, 59, 61, 62, 62, 65, 66, 69, 69, 70, 74, 74, 75, 77, 77, 78, 81, 83, 84, 86, 86, 89, 89, 90, 90, 91, 93, 94, 94, 98, 98, 101, 101, 101, 104, 105, 105, 106, 107, 109, 110, 110, 110, 113, 114, 115, 116, 117

Offset: 1

Clark Kimberling

Numbers n = x^2 + y^2 + z^2, with 0

Records of 1,2,3 etc. repetitions are set by the numbers 14, 62, 101, 161, 206, 314, 341, 446, 689, 734, 854, 1106, 1154, 1286, 1454 etc.; see A025415. - R. J. Mathar, Mar 15 2007

			14 = 1^2 + 2^2 + 3^2.
62 = 1^2 + 5^2 + 6^2 = 2^2 + 3^2 + 7^2.
105 = 1^2 + 2^2 + 10^2 = 4^2 + 5^2 + 8^2.
122 = 3^2 + 7^2 + 8^2 = 4^2 + 5^2 + 9^2.

Georg Fischer, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Cf. A004432, A001974.

A024803 := proc(n) local a,x,y,z ; a := 0 ; for x from 1 to floor(sqrt(n)) do for y from x+1 to floor(sqrt(n-x^2)) do z := n-x^2-y^2 ; if issqr(z) then z := sqrt(z) ; if z>y and z>x then a := a+1 ; fi ; fi ; od ; od ; RETURN(a) ; end: for n from 1 to 200 do a := A024803(n) : for i from 1 to a do printf("%d ",n) ; od ; od : # R. J. Mathar, Mar 15 2007

Zak Seidov pointed out that there were errors. These have now been corrected. - N. J. A. Sloane, Dec 05 2006

More terms from R. J. Mathar, Mar 15 2007

A374227 a(n) is the smallest number which can be represented as the sum of three distinct positive n-th powers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

6, 62, 5104, 5978882

Offset: 1

Ilya Gutkovskiy, Jul 01 2024

			a(3) = 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.

Cf. A025415, A025419, A123075, A146756, A374226, A374228.

A214512 Least number having n orderless representations as p^2 + q^2 + r^2, where p, q, and r are primes.

Original entry on oeis.org

12, 219, 363, 699, 1179, 2019, 2259, 3891, 4059, 6459, 5379, 10899, 13179, 10659, 12579, 21819, 20979, 26859, 34419, 38379, 41019, 61299, 39459, 41811, 82131, 50379, 77451, 71379, 141099, 85491, 103971, 74571, 180411, 108339, 179739, 161139, 126819, 225099

Offset: 1

T. D. Noe, Jul 29 2012

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A025414, A025415, A214511, A214513.

nn = 10^6; ps = Prime[Range[PrimePi[Sqrt[nn]]]]; t = Flatten[Table[ps[[i]]^2 + ps[[j]]^2 + ps[[k]]^2, {i, Length[ps]}, {j, i, Length[ps]}, {k, j, Length[ps]}]]; t = Select[t, # <= nn &]; t2 = Sort[Tally[t]]; u = Union[Transpose[t2][[2]]]; d = Complement[Range[u[[-1]]], u]; If[d == {}, nLim = u[[-1]], nLim = d[[1]]-1]; t3 = Table[Select[t2, #[[2]] == n &, 1][[1]], {n, nLim}]; Transpose[t3][[1]]

A374806 a(n) is the smallest number which can be represented as the sum of 3 distinct nonzero triangular numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

10, 19, 37, 52, 82, 109, 136, 241, 226, 217, 247, 364, 427, 457, 541, 532, 577, 637, 961, 721, 787, 1066, 1102, 1381, 1267, 1564, 1192, 1396, 1816, 1501, 1612, 1927, 1942, 2242, 1792, 2842, 2587, 2557, 2422, 2866, 2887, 3181, 3271, 3412, 4126

Offset: 1

Ilya Gutkovskiy, Jul 20 2024

nonn

			a(3) = 37 = 1 + 15 + 21 = 3 + 6 + 28 = 6 + 10 + 21.

Eric Weisstein's World of Mathematics, Triangular Number

Cf. A000217, A025415, A342326, A374807.

A374273 a(n) is the smallest number which can be represented as the sum of three distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

37, 161, 498, 1666, 2546, 7434, 16609, 25952, 48786, 49861, 72347, 127335, 183289, 196469, 416913, 466546, 494369, 506649, 801010, 1401011, 2372586, 1414009, 2003027, 3274986, 2927260, 2721677, 5592756, 8016592, 6632759, 7057914, 8401837, 13248146, 11648679, 8650006

Offset: 3

Ilya Gutkovskiy, Jul 02 2024

nonn

			a(3) = 37 = 1 + 15 + 21 = 3 + 6 + 28 = 6 + 10 + 21.
a(4) = 161 = 1^2 + 4^2 + 12^2 = 2^2 + 6^2 + 11^2 = 4^2 + 8^2 + 9^2 = 5^2 + 6^2 + 10^2.

Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A025415, A057145, A350405, A374144, A374274.

a(21) and beyond from Michael S. Branicky, Jul 08 2024

A374693 a(n) is the smallest number which can be represented as the sum of 3 distinct nonzero fourth powers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

98, 6578, 811538, 5978882, 1289202642, 292965218, 779888018, 5745705602, 105760443698, 49511121842, 1872511131218, 281539574498, 17673688436978

Offset: 1

Ilya Gutkovskiy, Jul 17 2024

a(16) = 7865870969138. - Michael S. Branicky, Jul 23 2024

			a(2) = 6578 = 1^4 + 2^4 + 9^4 = 3^4 + 7^4 + 8^4.
a(3) = 811538 = 4^4 + 23^4 + 27^4 = 7^4 + 21^4 + 28^4 = 12^4 + 17^4 + 29^4.

Cf. A025415, A025419, A230562, A374696.

a(9)-a(12) from Michael S. Branicky, Jul 22 2024

a(13) from Michael S. Branicky, Jul 23 2024

A374805 a(n) is the smallest positive integer whose square can be represented as the sum of 3 distinct nonzero squares in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

1, 7, 15, 23, 31, 21, 33, 39, 49, 45, 79, 57, 95, 103, 75, 69, 127, 87, 63, 151, 93, 167, 111, 123, 99, 187, 117, 105, 161, 241, 141, 135, 153, 247, 271, 283, 177, 183, 165, 275, 147, 171, 323, 219

Offset: 0

Ilya Gutkovskiy, Jul 20 2024

			a(3) = 23: 23^2 = 3^2 + 6^2 + 22^2 = 3^2 + 14^2 + 18^2 = 6^2 + 13^2 + 18^2.

Index entries for sequences related to sums of squares

Cf. A006339, A025415, A025442, A346071.

a(35)-a(43) from Michael S. Branicky, Jul 21 2024

A096021 Greatest number, not divisible by 4, having exactly n partitions into three distinct positive squares.

Original entry on oeis.org

1507, 4323, 5947, 10707, 19723, 30067, 34483, 47107, 58843, 77683, 111763, 106723, 126043, 166147, 164803, 222643, 217627, 232243, 289963, 319243, 300787, 319867, 462883, 393187, 546067, 532123, 502483, 615883, 662803, 606643

Offset: 1

T. D. Noe, Jun 15 2004

nonn

These are conjectured values. The Mathematica program checks numbers up to 10^6.

Cf. A025415 (least sum of 3 distinct nonzero squares in exactly n ways).

lim=1000; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && n0&]], {k, 30}]

Showing 1-9 of 9 results.

cp's OEIS Frontend

A350241 a(n) is the smallest number which can be represented as the sum of n distinct nonzero squares in exactly n ways, or 0 if no such number exists.

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A024803 Numbers that are the sum of 3 distinct nonzero squares, including repetitions.

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A374227 a(n) is the smallest number which can be represented as the sum of three distinct positive n-th powers in exactly n ways, or -1 if no such number exists.

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A214512 Least number having n orderless representations as p^2 + q^2 + r^2, where p, q, and r are primes.

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Mathematica

A374806 a(n) is the smallest number which can be represented as the sum of 3 distinct nonzero triangular numbers in exactly n ways, or -1 if no such number exists.

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A374273 a(n) is the smallest number which can be represented as the sum of three distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

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A374693 a(n) is the smallest number which can be represented as the sum of 3 distinct nonzero fourth powers in exactly n ways, or -1 if no such number exists.

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Extensions

A374805 a(n) is the smallest positive integer whose square can be represented as the sum of 3 distinct nonzero squares in exactly n ways, or -1 if no such number exists.

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Extensions

A096021 Greatest number, not divisible by 4, having exactly n partitions into three distinct positive squares.

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Author

Keywords

Comments

Crossrefs

Programs

Mathematica