A101506 -id:A101506 - cp's OEIS frontend

A101507 Numbers n such that exp(n) has a smaller relative error abs(exp(n)/m!-1) in approximating the closest factorial m!>1 than exp(k) for any k with 1

2, 3, 15, 20, 58, 2893, 3172, 13778, 36596, 63894, 208744, 296557, 404667, 11500740, 17800369, 37858613, 38393813, 902477623, 4126573365, 79491128275, 338814192247, 1599109448865

Offset: 1

Hugo Pfoertner, Dec 20 2004

Numbers n such that abs(exp(n)/m!-1)

			a(1)=2 because exp(2)=7.389 is a better approximation to the nearest factorial 3!=6 with +23% relative error than is exp(1)=2.718 for its closest factorial 2!=2 with +36% relative error.
a(2)=3: exp(3)/4!-1=-0.1631. The next improvement occurs for a(3)=15 because exp(15)/10!-1=-0.099.
a(22)=1599109448865: The relative error of exp(1599109448865) in approximating A101506(22)!=66836971558! is 1.276*10^(-12).

Cf. A101506.

A241730 Number of digits in the decimal expansion of n! minus the number of digits in the decimal expansion of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 39, 40, 42, 43, 45, 46, 48, 50, 51, 53, 55, 56, 58, 60, 61, 63, 65, 66, 68, 70, 72, 73, 75, 77, 79, 80, 82, 84, 86, 88, 89, 91, 93, 95, 97, 99, 100, 102, 104, 106

Offset: 0

Alonso del Arte, Apr 27 2014

			a(5) = 2 because 5! = 120 has three digits, 5 has only one digit, and 3 - 1 = 2.
a(6) = 2 because 6! = 720 has three digits, 6 has only one digit, and 3 - 1 = 2.
a(7) = 3 because 7! = 5040 has four digits, 7 has only one digit, and 4 - 1 = 3.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A034886, A101506.

seq(nops(convert(n!,base,10)) - nops(convert(n,base,10)), n=0..50); # Peter Luschny, Apr 28 2014

Table[Length[IntegerDigits[n!]] - Length[IntegerDigits[n]], {n, 0, 79}]

a(n)=lngamma(n+1)\log(10)-#Str(n)+1 \\ Charles R Greathouse IV, Apr 28 2014

a(n) = floor(log(n!)/log(10)) - floor(log(n)/log(10)) for n > 0.

a(n) = A034886(n) - A055642(n).

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A101507 Numbers n such that exp(n) has a smaller relative error abs(exp(n)/m!-1) in approximating the closest factorial m!>1 than exp(k) for any k with 1

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A241730 Number of digits in the decimal expansion of n! minus the number of digits in the decimal expansion of n.

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