A098475 -id:A098475 - cp's OEIS frontend

A158661 Least number k such that sigma_n(k) > sigma_n(k+1), where sigma_n(k) = sum of the n-th powers of the divisors of k.

4, 4, 6, 24, 60, 144, 360, 852, 1968, 4488, 10068, 22272, 48780, 105948, 228588, 490404, 1046976, 2225964, 4715400, 9956976, 20965212, 44031360, 92262348, 192920784, 402629256, 838827576, 1744784388, 3623814864, 7516104564

Offset: 0

T. D. Noe, Mar 23 2009

nonn

It appears that the inequality a(n+1) > (2+2/n)*a(n) is true for n > 4.

			The values of the sigma_3 function (A001158) are increasing up to 25. Hence a(3)=24.

Join[{4,4}, Table[k=Floor[NSolve[Zeta[n](x-1)^n==x^n, x, WorkingPrecision->100][[ -1,1,2]]]; While[DivisorSigma[n,k]

For n>0, a(n) = A098475(n) - 1.

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

Original entry on oeis.org

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

Matthew Conroy, Mar 04 2025

nonn

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.

			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.

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

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;

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

cp's OEIS Frontend

A158661 Least number k such that sigma_n(k) > sigma_n(k+1), where sigma_n(k) = sum of the n-th powers of the divisors of k.

Views

Author

Keywords

Comments

Examples

Programs

Mathematica

Formula