A064876 -id:A064876 - cp's OEIS frontend

A064873 First of four sequences representing the lexicographical minimal decomposition of n in 4 squares: n = a(n)^2 + A064874(n)^2 + A064875(n)^2 + A064876(n)^2.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 0

Reinhard Zumkeller, Oct 10 2001

nonn

A072401(n) = A057427(a(n)).

For k<112: a(n)=A072401(n), but A072401(112) = 1<>a(112)=2, as also A072401(112 - 1) = 1.

			a(25) = 0: 25 = a(25)^2 + A064874(25)^2 + A064875(25)^2 + A064876(25)^2 = 0 + 0 + 0 + 25 and the other decompositions (0, 0, 3, 4) and (1, 2, 2, 4) are greater than (0, 0, 0, 5).

Eric Weisstein's World of Mathematics, Square Numbers.
Index entries for sequences related to sums of squares.

Cf. A064874, A064875, A064876, A071374.

A064874 Second of four sequences representing the lexicographical minimal decomposition of n in 4 squares: n = A064873(n)^2 + a(n)^2 + A064875(n)^2 + A064876(n)^2.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 2, 0, 1, 1, 0, 0, 0, 1, 0, 1, 2, 2, 2, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 3, 2, 0, 1, 1, 4, 0, 0, 1, 0, 0, 1, 1, 2, 2, 0, 1, 1, 0, 1, 1, 0, 0, 1, 3, 0, 1, 3, 3, 0, 0, 0, 1, 2, 2, 2, 2, 0, 0, 0, 1, 2, 0, 1, 1, 4, 0, 0, 1, 1, 2, 2, 2, 4, 0, 0, 1, 0, 0, 1, 1, 0

Offset: 0

Reinhard Zumkeller, Oct 10 2001

nonn

			a(19) = 1: 19 = A064873(19)^2 + a(19)^2 + A064875(19)^2 + A064876(19)^2 = 0 + 1 + 9 + 9 and the other decomposition (1, 1, 1, 4) is greater than (0, 1, 3, 3).

Cf. A064873, A064875, A064876, A064877.

A064875 Third of four sequences representing the lexicographical minimal decomposition of n in four squares: n = A064873(n)^2 + A064874(n)^2 + a(n)^2 + A064876(n)^2.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 1, 1, 2, 0, 1, 1, 2, 2, 2, 2, 0, 1, 3, 3, 2, 2, 3, 3, 2, 0, 1, 1, 1, 2, 2, 2, 4, 4, 3, 3, 0, 1, 1, 1, 2, 4, 4, 3, 2, 3, 3, 3, 4, 0, 1, 1, 4, 2, 2, 2, 4, 2, 3, 3, 3, 5, 5, 5, 0, 1, 1, 3, 2, 2, 5, 5, 6, 3, 5, 5, 6, 3, 5, 5, 4, 0, 1, 1, 4, 2, 2, 2, 6, 5, 3, 3, 3, 5, 3, 3, 4, 4, 7, 7, 0, 1, 1, 1, 2

Offset: 0

Reinhard Zumkeller, Oct 10 2001

nonn

			a(19) = 3: 19 = A064873(19)^2 + A064874(19)^2 + a(19)^2 + A064876(19)^2 = 0 + 1 + 9 + 9 and the other decomposition (1, 1, 1, 4) is greater than (0, 1, 3, 3).

Cf. A064873, A064874, A064876, A064877.

A064877 a(n) = n - (A064873(n) + A064874(n) + A064875(n) + A064876(n)).

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 2, 2, 4, 6, 6, 6, 6, 8, 8, 8, 12, 12, 12, 12, 14, 14, 14, 14, 16, 20, 20, 20, 20, 22, 22, 22, 24, 24, 26, 26, 30, 30, 30, 30, 32, 32, 32, 32, 34, 36, 36, 36, 36, 42, 42, 42, 42, 44, 44, 44, 44, 46, 48, 48, 48, 50, 50, 50, 56, 56, 56, 54, 58, 58, 56, 56, 60, 62

Offset: 0

Reinhard Zumkeller, Oct 10 2001

nonn

Cf. A064873, A064874, A064875, A064876.

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A064873 First of four sequences representing the lexicographical minimal decomposition of n in 4 squares: n = a(n)^2 + A064874(n)^2 + A064875(n)^2 + A064876(n)^2.

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A064874 Second of four sequences representing the lexicographical minimal decomposition of n in 4 squares: n = A064873(n)^2 + a(n)^2 + A064875(n)^2 + A064876(n)^2.

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A064875 Third of four sequences representing the lexicographical minimal decomposition of n in four squares: n = A064873(n)^2 + A064874(n)^2 + a(n)^2 + A064876(n)^2.

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A064877 a(n) = n - (A064873(n) + A064874(n) + A064875(n) + A064876(n)).

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

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