cp's OEIS Frontend

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

%I A381708 #42 Mar 25 2025 20:24:52
%S A381708 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,
%T A381708 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,
%U A381708 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
%N A381708 a(n) is the smallest nonnegative integer k such that sigma_k(n) > sigma_k(j) for all 1 <= j < n.
%C A381708 sigma_k(n) is the sum of the k-th powers of the divisors of n.
%C A381708 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.
%F A381708 a(n) = 0 precisely when n is highly composite number A002182.
%F A381708 a(n) = 1 precisely when n is highly abundant A002093 and not highly composite.
%F A381708 a(n) = 2 precisely when n is in A193988 and is not highly composite and is not highly abundant.
%F A381708 a(n) <= m if n < A098475(m). Empirically, it appears that a(A098475(m)) = m+1. - _Pontus von Brömssen_, Mar 16 2025
%e A381708 For n = 1, k = 0 is enough so a(1) = 0.
%e A381708 For n = 2, k = 0 works since sigma_0(2) = 2 > 1 = sigma_0(1) so a(2) = 0.
%e A381708 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.
%e A381708 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.
%e A381708 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.
%o A381708 (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));
%o A381708 a(n) = my(ii=0); while(!check(n, ii), ii++);  ii;
%Y A381708 Cf. A000203, A001157, A002093, A002182, A098475, A109974, A193988.
%K A381708 nonn
%O A381708 1,5
%A A381708 _Matthew Conroy_, Mar 04 2025