A365108 a(n) is the smallest integer value of (p^n - q^n)/n for all choices of integers p > q >= 0.

1, 2, 9, 4, 625, 672, 117649, 32, 2187, 5941760, 25937424601, 1397760, 23298085122481, 308548739072, 29192926025390625, 4096, 48661191875666868481, 3817734144, 104127350297911241532841, 174339220, 209430786243, 24639156314201655345152, 907846434775996175406740561329

Offset: 1

Author

Felix Huber, Aug 21 2023

Keywords

• nonn

Comments

(p^n - q^n)/n has the integer value n^(n - 1) for p = n and q = 0. For p = n + k + 1 (k: nonnegative integer) the term has its minimum for q = n + k. With the binomial theorem follows ((n + k + 1)^n - (n + k)^n)/n >= ((n + k)^n - n*(n + k)^(n - 1) - (n + k)^n)/n = (n + k)^(n - 1) >= n^(n - 1). Therefore, for p > n, there is no smaller value of (p^n - q^n)/n than n^(n - 1). Thus a(n) <= n^(n - 1) exists with 1 <= p <= n and 0 <= q <= p - 1.

a(n) is also the smallest integer value that the integral over f(x) = x^(n - 1) between the nonnegative integer integration limits q and p (p > q) can have.

Examples

			For n = 5, a(5) = 672 with p = 4 and q = 2.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..388

Crossrefs

Cf. A000169, A055860, A020725.

Programs

Maple

A365108 := proc(n) local q, p, s, a_n; a_n := n^(n - 1); for p to n do for q from 0 to p - 1 do s := (p^n - q^n)/n; if s = floor(s) and s < a_n then a_n := s; end if; end do; end do; return a_n; end proc;
seq(A365108(n), n = 1 .. 23);

Python

from sympy.ntheory.residue_ntheory import nthroot_mod
def A365108(n):
    c, qdict = n**(n-1), {}
    for p in range(1,n+1):
        r, m = pow(p,n,n), p**n
        if r not in qdict:
            qdict[r] = tuple(nthroot_mod(r,n,n,all_roots=True))
        c = min(c,min(((m-q**n)//n for q in qdict[r] if qChai Wah Wu, Sep 23 2023

Formula

a(n) is the integer minimum of (p^n - q^n)/n for 1 <= p <= n and 0 <= q <= p - 1.