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

1,2

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.

Links

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

Formula

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

Example

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

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);

Prog

(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 q<p), default=c))
    return int(c) # Chai Wah Wu, Sep 23 2023

Crossrefs

Cf. A000169, A055860, A020725.

Sequence in context: A003725 A292952 A328619 * A076930 A309926 A011068

Adjacent sequences: A365105 A365106 A365107 * A365109 A365110 A365111

Keyword

nonn

Author

Felix Huber, Aug 21 2023

Status

approved