A064873 -id:A064873 - cp's OEIS frontend

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.

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.

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

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, 4, 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, 6, 6, 8, 8, 8, 7, 8, 8, 6, 6, 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, 7, 10, 10, 10

Offset: 0

Reinhard Zumkeller, Oct 10 2001

nonn

			a(18) = 3: 18 = A064873(18)^2 + A064874(18)^2 + A064875(18)^2 + a(18)^2 = 0 + 0 + 9 + 9 and the other decompositions (0, 1, 1, 4) and (1, 2, 2, 3) are greater than (0, 0, 3, 3).

Cf. A064873, A064874, A064875, 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.

A072401 1 iff n is of the form 4^m*(8k+7).

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Jun 16 2002

Characteristic function of A004215, indicating numbers not the sum of 3 integer squares.

a(n) + 1 is the smallest positive number such that (a(n) + 1) * n is the sum of three squares. - Peter Schorn, Jul 18 2023

Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197; preprint. See Example 20.
Eric Weisstein's World of Mathematics, Square Numbers.
Index entries for sequences related to sums of squares.
Index entries for characteristic functions.

Cf. A004215, A057427, A064873, A072400.

Cf. A071374.

A072400[n_] := Mod[If[Mod[n, 4] == 0, n/4^IntegerExponent[n, 4], n], 8];
a[n_] := 1 - Sign[7 - A072400[n]];
Table[a[n], {n, 0, 96}] (* Jean-François Alcover, Dec 13 2021 *)

a(n) = if(n, (n >> (2*valuation(n, 4))) % 8 == 7, 0); \\ Amiram Eldar, May 15 2025

def A072401(n): return ((m:=(~n&n-1).bit_length())&1^1)&int((n>>m)&7==7) # Chai Wah Wu, Aug 01 2023

a(n) = 1 - A057427(7 - A072400(n)).
a(A004215(k)) = 1 for k>0.
a(n) = A057427(A064873(n)).
For n<112: a(n) = A064873(n), but A064873(112) = 2, as also a(112 - 1) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/6. - Amiram Eldar, May 15 2025

cp's OEIS Frontend

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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A064876 Last of four sequences representing the lexicographical minimal decomposition of n in 4 squares: n = A064873(n)^2 + A064874(n)^2 + A064875(n)^2 + a(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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A072401 1 iff n is of the form 4^m*(8k+7).

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