A368366 AGM transform of positive integers (see Comments for definition).

0, 1, 54, 3856, 384375, 52173801, 9342271792, 2144652558336, 616093495529805, 217007162119140625, 92121505246667356416, 46444033776765696086016, 27459259766085858672714571, 18830590227539089561714381425, 14834398958231516437500000000000

Offset: 1

Author

N. J. A. Sloane, Jan 24 2024

Keywords

• nonn

Comments

The AGM transform {AGM(n): n >= 1} is a measure of the difference between the arithmetic mean A(n) = S(n)/n and the geometric mean G(n) = P(n)^(1/n) of a sequence {a(n): n >= 1}, where S(n) = a(1)+...+a(n), P(n) = a(1)*...*a(n). It is given by AGM(n) = S(n)^n - n^n*P(n).

For odd n, these terms appear to be divisible by n^n; for even n, by (n/2)^n. Additional reductions may be possible. For example, with n = 7, 11, 15, 19, ..., 59, the terms are also divisible by these powers of two: 4, 8, 11, 16, 19, 23, 26, 32, 35, 39, 42, 47, 50, 54. - Hans Havermann, Jan 24 2024

Since a(n) = n^n*(((n+1)/2)^n-n!) = (n(n+1)/2)^n-n^n*n!, a(n) is divisible by n^n for odd n and divisible by (n/2)^n for even n. - Chai Wah Wu, Jan 25 2024

Links

• Paolo Xausa, Table of n, a(n) for n = 1..220

Crossrefs

Cf. A000027, A127610, A368372/A368373.

See A368367-A368371, A369394 for further examples.

The AGM transform of (n mod 2) is A276978.

A368374 gives another way to look at the problem.

Programs

Maple

AGM := proc(f,M) local b,n,S,P,i,t; b:=[];
for n from 1 to M do
S:=add(f(i),i=1..n); P:=mul(f(i),i=1..n); t:=S^n-n^n*P;
b:=[op(b),t];
od:
b;
end;
fid:=proc(n) n; end; # the identity map
AGM(fid,20);

Mathematica

A368366[n_] := n^n (((n + 1)/2)^n - n!);
Array[A368366, 10] (* Paolo Xausa, Jan 29 2024 *)

PARI

a368366(n) = {my(v=vector(n,i,i)); vecsum(v)^n - n^n*vecprod(v)}; \\ Hugo Pfoertner, Jan 24 2024

Python

from itertools import count, islice
def AGM(g): # generator of AGM transform of sequence given by generator g
    S, P = 0, 1
    for n, an in enumerate(g, 1):
        S += an
        P *= an
        yield S**n-n**n*P
print(list(islice(AGM(count(1)), 15))) # Michael S. Branicky, Jan 24 2024

Python

from math import factorial
def A368366(n): return ((m:=n**n)*(n+1)**n>>n)-m*factorial(n) # Chai Wah Wu, Jan 25 2024