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

1,3

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

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 *)

Prog

(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

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.

Sequence in context: A212705 A046199 A245831 * A291071 A254620 A299862

Adjacent sequences: A368363 A368364 A368365 * A368367 A368368 A368369

Keyword

nonn

Author

N. J. A. Sloane, Jan 24 2024

Status

approved