A020725 -id:A020725 - cp's OEIS frontend

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

Felix Huber, Aug 21 2023

nonn

(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.

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

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

Cf. A000169, A055860, A020725.

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);

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

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

A157319 Possible total points for a single team in a game of American football, ignoring safeties (and time constraints).

Original entry on oeis.org

0, 3, 6, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Dan Brown (ddbhockey(AT)hotmail.com), Feb 26 2009

Allowing 2-point safeties, the only impossible score is 1, see A087156. (With 1-point safeties all scores are possible.) - Charles R Greathouse IV, Jun 10 2015

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A069745, A020725.

a(n)=if(n<7,[0, 3, 6, 7, 9, 10][n],n+5) \\ Charles R Greathouse IV, Sep 02 2011

All combinations of sums of multiples of 3, 6, and 7.

Edited by N. J. A. Sloane, Mar 08 2009

A179870 a(n) = ((n-1)! * (n+1)!) ^ 2.

Original entry on oeis.org

4, 36, 2304, 518400, 298598400, 365783040000, 842764124160000, 3344930808791040000, 21407557176262656000000, 209815467884550291456000000, 3021342737537524196966400000000, 61783437639904832303765913600000000

Offset: 1

Jaroslav Krizek, Jul 30 2010

Table[((n-1)!(n+1)!)^2,{n,15}] (* Harvey P. Dale, Jul 12 2020 *)

a(n)=(n-1)!^4*(n^4+2*n^3+n^2) \\ Charles R Greathouse IV, Jan 03 2013

a(n) = A175430(n) ^ 2 = ((Product_(k=1,2,...,n) k*A020725(k)) / n) ^ 2 = ((Product_(k=1,2,...,n) k*(k+1)) / n) ^ 2.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

Formula

A157319 Possible total points for a single team in a game of American football, ignoring safeties (and time constraints).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A179870 a(n) = ((n-1)! * (n+1)!) ^ 2.

Views

Author

Keywords

Programs

Mathematica

PARI

Formula