A053867 Parity of sum of divisors of n less than n.

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

Offset: 1

Author

Henry Bottomley, Mar 29 2000

Keywords

• nonn
• easy

Comments

Partial sums up to n is about n/2. - David A. Corneth, Oct 20 2017

Examples

			a(9) = 0 because sum of proper divisors of 9 is 1 + 3 = 4 which is an even number.
From _David A. Corneth_, Oct 20 2017: (Start)
a(25) = 0 because 25 is an odd square. Therefore, a(2*25) = a(50) = 1 - a(25) = 1 and a(100) = a(2*50) = 1.
a(27) = 1 because 17 isn't an odd square. Therefore, a(2*27) = a(54) = 1-a(27) = 0 and a(108) = a(2*54) = 0. (End)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for characteristic functions.
• Index entries for sequences related to sums of divisors.

Crossrefs

Cf. A000005, A000265, A001065, A010052, A053866.

Programs

Mathematica

a[n_] = Mod[DivisorSigma[1, n] - n, 2]; Array[a, 100] (* Amiram Eldar, Jan 30 2025 *)

PARI

A053867(n) = ((sigma(n)-n)%2); \\ Antti Karttunen, Oct 20 2017

PARI

first(n) = my(res = vector(n, i, i%2)); forstep(i=1,sqrtint(n),2, for(j=0,logint(n\i^2,2), c = i^2 << j; res[c] = 1 - res[c])); res \\ David A. Corneth, Oct 20 2017

Formula

a(n) = A001065(n) mod 2.

a(2n+1) = 1 - A010052(2n+1); a(4n + 2) = 1 - a(2n + 1); a(4n) = a(2n). - David A. Corneth, Oct 20 2017

From Amiram Eldar, Jan 30 2025: (Start)

a(n) = 0 if n is an odd square, 1 is n is either an even square or twice a square, and n mod 2 otherwise.

Sum_{k=1..n} a(k) = n/2 + sqrt(n/2) + o(sqrt(n)). (End)

Extensions

More terms from James Sellers, Apr 08 2000