A054382 James Joyce's "Ulysses" sequence: number of digits in n^(n^n).

1, 1, 2, 13, 155, 2185, 36306, 695975, 15151336, 369693100, 10000000001, 297121486765, 9622088391635, 337385711567665, 12735782555419983, 515003176870815368, 22212093154093428530, 1017876887958723919835, 49390464231494436119285

Offset: 0

Author

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 07 2000

Keywords

• nonn
• base

Comments

Although Joyce mentions (9^9)^9, he clearly intended to refer to 9^(9^9).

(9^9)^9 is only 196627050475552913618075908526912116283103450944214766927315415537966391196809, whereas 9^(9^9) has 369693100 digits.

Examples

			"Because some years previously in 1886 when occupied with the problem of the quadrature of the circle he had learned of the existence of a number computed to a relative degree of accuracy to be of such magnitude and of so many places, e.g., the 9th power of the 9th power of 9, that, the result having been obtained, 33 closely printed volumes of 1000 pages each of innumerable quires and reams of India paper would have to be requisitioned in order to contain the complete tale of its printed integers of units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, hundreds of millions, billions, the nucleus of the nebula of every digit of every series containing succinctly the potentiality of being raised to the utmost kinetic elaboration of any power of any of its powers." - James Joyce, Ulysses, Chapter 17.
a(2)=2 since 2^(2^2)=2^4=16 has 2 digits. - _Carmine Suriano_, Feb 01 2011
a(0)=1 because 0^(0^0)=0^1=0, which has 1 digit. - _T. D. Noe_, Feb 01 2011
		

References

• C. A. Laisant (1906) proved that the number of digits of a(9), 9^9^9, is 369693100. H. S. Uhler (1947) published the log of the number to 250 decimal places.
• David Wells: The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, 1986, p. 208.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..1000 (first 57 terms from Carmine Suriano)
• H. Havermann, 9^9^9, in 33 volumes.
• J. Joyce, Ulysses, Ithaca chapter
• Eric Weisstein's World of Mathematics, Joyce Sequence.
• Eric Weisstein's World of Mathematics, Sultan's Dowry Problem.

Crossrefs

Cf. A002488, A055642, A066022.

Programs

Maple

A055642 := proc(n) max(1,ilog10(n)+1) ; end proc:
A054382 := proc(n) A055642(n^(n^n)) ; end proc: # R. J. Mathar, Feb 01 2011

Mathematica

f[ j_ ] := 1 + Floor[ Log[10, j] j^j ]; Table[ f[j], {j, 2, 20} ]

Formula

a(n) = floor(n^n*log_10(n)) + 1 for n > 0. - Jianing Song, Nov 21 2018

Extensions

More terms from Michael Kleber, May 07 2000