A381708 a(n) is the smallest nonnegative integer k such that sigma_k(n) > sigma_k(j) for all 1 <= j < n.

0, 0, 1, 0, 2, 0, 3, 1, 2, 1, 3, 0, 3, 2, 2, 1, 3, 1, 3, 1, 3, 2, 3, 0, 4, 3, 3, 2, 4, 1, 4, 2, 4, 2, 4, 0, 4, 3, 4, 2, 4, 1, 4, 2, 4, 2, 4, 0, 4, 3, 4, 2, 4, 2, 4, 2, 4, 3, 4, 0, 5, 3, 4, 2, 5, 2, 5, 3, 4, 2, 5, 1, 5, 3, 4, 3, 5, 2, 5, 2, 5, 3, 5, 1, 5, 3, 5, 3, 5, 1, 5, 3, 5, 3, 5, 1, 5, 3, 5, 2, 5, 2, 5, 3, 5, 3, 5, 1, 5

Offset: 1

Author

Matthew Conroy, Mar 04 2025

Keywords

• nonn

Comments

sigma_k(n) is the sum of the k-th powers of the divisors of n.

a(n) exists since one can prove that for k > n*(log 2 + 1/2 log(n-1)), sigma_k sets a record at n.

Examples

			For n = 1, k = 0 is enough so a(1) = 0.
For n = 2, k = 0 works since sigma_0(2) = 2 > 1 = sigma_0(1) so a(2) = 0.
For n = 3, sigma_0(3) = 2 = sigma_0(2), but sigma_1(3) = 1^1+3^1 = 4 > 3 = sigma_1(2) > 1 = sigma_1(1) so a(3) = 1.
For n = 4, sigma_0(4) = 1^0+2^0+4^0 = 3 > 2 = sigma_0(3) = sigma_0(2) > 1 = sigma_0(1) so a(4) = 0.
For n = 5, sigma_0(5) = 2 = sigma_0(2) and sigma_1(5) = 6 < sigma_1(4) = 7 but sigma_2(5) = 26 > sigma_2(4) > sigma_2(3) > sigma_2(2) > sigma_2(1) so a(5) = 2.
		

Crossrefs

Cf. A000203, A001157, A002093, A002182, A098475, A109974, A193988.

Programs

PARI

check(n,k) =  my(m=0);for(i=1,n-1, my(s=sigma(i,k)); if(s>m,m=s)); if(sigma(n,k)>m,return(1),return(0));
a(n) = my(ii=0); while(!check(n, ii), ii++);  ii;

Formula

a(n) = 0 precisely when n is highly composite number A002182.

a(n) = 1 precisely when n is highly abundant A002093 and not highly composite.

a(n) = 2 precisely when n is in A193988 and is not highly composite and is not highly abundant.

a(n) <= m if n < A098475(m). Empirically, it appears that a(A098475(m)) = m+1. - Pontus von Brömssen, Mar 16 2025