A180466 -id:A180466 - cp's OEIS frontend

A262689 a(n) = largest number k <= A000196(n) for which A002828(n-(k^2)) = A002828(n)-1.

0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 4, 4, 3, 3, 4, 4, 3, 3, 4, 5, 5, 5, 5, 5, 5, 5, 4, 5, 5, 5, 6, 6, 6, 6, 6, 5, 5, 5, 6, 6, 6, 6, 4, 7, 7, 7, 6, 7, 7, 7, 6, 7, 7, 7, 7, 6, 7, 7, 8, 8, 8, 7, 8, 8, 6, 7, 6, 8, 7, 7, 6, 8, 7, 7, 8, 9, 9, 9, 8, 9, 9, 9, 6, 8, 9, 9, 9, 8, 9, 9, 8, 9, 7, 9, 10, 10, 10, 10, 10, 10, 9, 9, 10, 10, 10, 10, 10, 8, 8, 9, 10, 9, 10, 10, 10, 11

Offset: 0

Antti Karttunen, Oct 03 2015

nonn

a(n) = square root of the largest summand present among all representations of n as a minimal number of squares, A002828(n). See the last two examples.

			For n = 9, we have A002828(9) = 1 because 9 is itself a perfect square. By the definition of this sequence, we find the largest k <= 3 for which A002828(9 - k^2) = A002828(9)-1 = 0, and it is k=3 that satisfies this condition, thus a(9) = 3.
For n = 27, by the other interpretation given in the Comments section, we see that the two minimal sums requiring the least number of squares (= 3 = A002828(27)) are (25 + 1 + 1) and (9 + 9 + 9). As 25 is larger than 9, we have a(27) = sqrt(25) = 5.
For n = 33, the two minimal solutions are (25 + 4 + 4) and (16 + 16 + 1). As 25 is larger than 16, we have a(33) = sqrt(25) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..65536

Cf. A000196, A002828, A010052, A064876, A180466, A262690.

Differs from A064876 for the first time at n=33, where a(33) = 5, while A064876(33) = 4.

Other identities. For all n >= 0:
a(n) = A000196(A262690(n)).
a(n^2) = n.

A258257 The number of representations of n as a minimal number of triangular numbers, A000217(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 2, 1, 2, 2, 1, 1, 1, 2, 5, 1, 2, 1, 2, 1, 1, 3, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 4, 2, 3, 1, 1, 2, 3, 6, 1, 1, 3, 3, 1

Offset: 1

Martin Renner, May 24 2015

nonn

			a(5) = 1 since 5 = 1 + 1 + 3 is the only representation as a minimal number of three triangular numbers.
a(16) = 2 since 16 = 1 + 15 = 6 + 10 has two representations as a minimal number of two triangular numbers.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A061336, A180466.

t[n_] := n (n + 1)/2; a[n_] := Block[{k = 1, t, tt = t /@ Range[ Sqrt[2*n]]}, While[{} == (r = IntegerPartitions[n, {k}, tt]), k++]; Length@r]; Array[a, 100] (* Giovanni Resta, Jun 09 2015 *)

A141490 Least number k having n representations as the sum of the minimal number of squares, A002828.

Original entry on oeis.org

1, 27, 28, 63, 103, 124, 135, 175, 207, 247, 255, 252, 327, 351, 412, 375, 511, 423, 543, 679, 540, 639, 687, 495, 567, 663, 759, 775, 847, 988, 783, 1111, 735, 1327, 855, 927, 1191, 999, 1308, 975, 1143, 1383, 1263, 1071, 1463, 1359, 1495, 1375, 1479

Offset: 1

Martin Renner, Jan 15 2011

nonn

That is, a(n) is the least k such that A180466(k) = n.

			a(1) = 1 since 1 = 1^2;
a(2) = 27 since 27 = 1^2 + 1^2 + 5^2 = 3^2 + 3^2 + 3^2 (2 ways);
a(3) = 28 since 28 = 1^2 + 1^2 + 1^2 +5^2 = 1^2 + 3^2 + 3^2 + 3^2 = 2^2 + 2^2 + 2^2 + 4^2 (3 ways).

Eric W. Weisstein, MathWorld: Waring's Problem

Cf. A180466 (number of representations of n as a minimal number of squares, A002828(n))

t=Table[r=PowersRepresentations[n, 4, 2]; Sort[Tally[4-Count[#, 0] & /@ r]][[1, 2]], {n, 1000}]; u=Union[t]; c=Complement[Range[Max[u]], u]; If[c == {}, mx=u[[-1]], mx=c[[1]]-1]; Flatten[Table[Position[t, n, 1, 1], {n, mx}]]

A258266 Numbers having only one representation as a sum of the minimal number of squares, A002828.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 32, 34, 35, 36, 37, 40, 41, 42, 43, 44, 45, 46, 48, 49, 52, 53, 56, 58, 61, 64, 67, 68, 70, 72, 73, 74, 76, 78, 80, 81, 82, 84, 88, 89, 90, 91, 93, 96, 97

Offset: 1

Martin Renner, May 25 2015

nonn

A180466(a(n)) = 1.

Complement of A258267.

Cf. A180466, A258264, A258267.

A258267 Numbers having more than one representation as a sum of the minimal number of squares, A002828.

Original entry on oeis.org

27, 28, 31, 33, 38, 39, 47, 50, 51, 54, 55, 57, 59, 60, 62, 63, 65, 66, 69, 71, 75, 77, 79, 83, 85, 86, 87, 92, 94, 95, 99, 102, 103, 105, 107, 108, 110, 111, 112, 114, 118, 119, 123, 124, 125, 126, 127, 129, 130, 131, 132, 134, 135, 138, 139, 141, 143, 145

Offset: 1

Martin Renner, May 25 2015

nonn

A180466(a(n)) > 1.

Complement of A258266.

Cf. A180466, A258265, A258266.

cp's OEIS Frontend

A262689 a(n) = largest number k <= A000196(n) for which A002828(n-(k^2)) = A002828(n)-1.

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A258257 The number of representations of n as a minimal number of triangular numbers, A000217(n).

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Mathematica

A141490 Least number k having n representations as the sum of the minimal number of squares, A002828.

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Programs

Mathematica

A258266 Numbers having only one representation as a sum of the minimal number of squares, A002828.

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A258267 Numbers having more than one representation as a sum of the minimal number of squares, A002828.

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