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Glaisher-Kinkelin Constant Digits


The decimal expansion of the Glaisher-Kinkelin constant is given by

 A=1.28242712...

(OEIS A074962). A was computed to 5×10^5 decimal digits by E. Weisstein (Dec. 3, 2015).

The Earls sequence (starting position of n copies of the digit n) for e is given for n=1, 2, ... by 7, 14, 2264, 1179, 411556, ... (OEIS A225763).

The digit sequences 0123456789 and 9876543210 do not occur in the first 5×10^5 digits (E. Weisstein, Dec. 3, 2015).

A-constant primes occur for 7, 10, 18, 64, 71, 527, 1992, 5644, 8813, 19692, ... (OEIS A118420) decimal digits.

The starting positions of the first occurrence of n=0, 1, 2, ... in the decimal expansion of A (including the initial 1 and counting it as the first digit) are 12, 1, 2, 18, 5, 22, 14, 7, 3, 10, ... (OEIS A229193).

Scanning the decimal expansion of A until all n-digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 5, 98, 478, 9192, ... (OEIS A000000), which end at digits 22, 495, 7233, 100426, ... (OEIS A000000).

It is not known if the Glaisher-Kinkelin constant is normal in base 10, but the following table giving the counts of digits in the first 10^n terms shows normal-appearing behavior up to at least 10^4

d\nOEIS1010010^310^410^5
0A000000011969999890
1A0000002910210339928
2A000000416939929977
3A00000008100101610055
4A000000189995510043
5A000000059497910034
6A0000000129698810121
7A00000011211410679998
8A00000011110810319999
9A00000018989409955

See also

Constant Digit Scanning, Constant Primes, Glaisher-Kinkelin Constant, Glaisher-Kinkelin Constant Continued Fraction

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References

Sloane, N. J. A. Sequences A074962, A118420, and A225763 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Glaisher-Kinkelin Constant Digits." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Glaisher-KinkelinConstantDigits.html

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