A072400 (Factors of 4 removed from n) modulo 8.

1, 2, 3, 1, 5, 6, 7, 2, 1, 2, 3, 3, 5, 6, 7, 1, 1, 2, 3, 5, 5, 6, 7, 6, 1, 2, 3, 7, 5, 6, 7, 2, 1, 2, 3, 1, 5, 6, 7, 2, 1, 2, 3, 3, 5, 6, 7, 3, 1, 2, 3, 5, 5, 6, 7, 6, 1, 2, 3, 7, 5, 6, 7, 1, 1, 2, 3, 1, 5, 6, 7, 2, 1, 2, 3, 3, 5, 6, 7, 5, 1, 2, 3, 5, 5, 6, 7, 6, 1, 2, 3, 7, 5, 6, 7, 6

Offset: 1

Author

Reinhard Zumkeller, Jun 16 2002

Keywords

• nonn
• easy

Comments

a(n) <> 7 iff n equals the sum of 3 integer squares.

a(A004215(k)) = 7 for k>0;

Examples

			From _Michael De Vlieger_, May 08 2017: (Start)
a(4) = 1 since 4 = 1 * 4^1 and 4 / 4^1 = 1; 1 = 1 (mod 8).
a(5) = 5 since it is not a multiple of 4; 5 = 5 (mod 8).
a(12) = 3 since 12 = 3 * 4^1 and 12 / 4^1 = 3; 3 = 3 (mod 8).
a(44) = 3 since 44 = 11 * 4^1 and 44 / 4^1 = 11; 3 = 11 (mod 8).
a(64) = 1 since 64 = 1 * 4^3 and 64 / 4^3 = 1; 1 = 1 (mod 8). (End)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Square Numbers.

Crossrefs

Cf. A000378, A004215, A057427, A065883, A072401, A286366.

Programs

Mathematica

Array[Mod[If[Mod[#, 4] == 0, #/4^IntegerExponent[#, 4], #], 8] &, 96] (* Michael De Vlieger, May 08 2017 *)

PARI

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

Python

def A072400(n): return (n>>((~n&n-1).bit_length()&-2))&7 # Chai Wah Wu, Aug 01 2023

Formula

a(n) = A065883(n) mod 8.

A072401(n) = 1 - A057427(7 - a(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4. - Amiram Eldar, May 15 2025

Extensions

Offset corrected (from 0 to 1) by Antti Karttunen, May 08 2017