A368374 a(n) = smallest k such that AM(k) - GM(k) >= n, where AM(k) and GM(k) are the arithmetic and geometric means of [1,...,k].

1, 11, 19, 27, 35, 43, 50, 58, 66, 74, 81, 89, 97, 104, 112, 120, 127, 135, 143, 150, 158, 165, 173, 181, 188, 196, 204, 211, 219, 226, 234, 242, 249, 257, 264, 272, 280, 287, 295, 302, 310, 318, 325, 333, 340, 348, 356, 363, 371, 378, 386, 394, 401, 409, 416

Offset: 0

Author

N. J. A. Sloane, Jan 27 2024, following a suggestion from Don Reble

Keywords

• nonn

Comments

The difference d(x) = AM(1,2,3,...,x) - GM(1,2,3,...,x) increases. The first difference of d(x) approaches a limit, 1/2 - 1/e (0.13212...). So we could define a(n) to be the least x such that d(x) >= n. - Don Reble, Jan 27 2024. Which is what I did.

Examples

			The values of AM(i)-GM(i) for i = 1, ..., 11 are 0, 0.0857864376269049512, 0.1828794071678603411, 0.2866361605993568152, 0.3948289153026481077, 0.5062048344760910451, 0.6199848408587035501, 0.7356494004968713999, 0.8528337256030871195, 0.9712713118832352378, 1.0907612204156046410, so a(1) = 11.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A068996, A368366.

Programs

Maple

Digits:=20;
AM := proc(n) local i; add(i,i=1..n)/n; end;
GM := proc(n) local i; mul(i,i=1..n)^(1/n); end;
don := proc(n) evalf(AM(n) - GM(n)); end;
a:=[1]; w:=1;
for i from 1 to 300 do
   if don(i) >= w then a:=[op(a),i]; w:=w+1; fi;
od:
a;

Python

from math import factorial
def A368374(n):
    if n == 0: return 1
    m = (n<<1)-1
    kmin, kmax = m, m
    while factorial(kmax)< (kmax-m)**kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if factorial(kmid)<Chai Wah Wu, Jan 27 2024

Extensions

a(39)-a(54) from Alois P. Heinz, Jan 27 2024