First digit counting compatibility for Niven integer powers

Werner Hürlimann


It is claimed that the first digits of Niven integer powers follow a generalized Benford law with a specific parameter-free size-dependent exponent that converges asymptotically to the inverse power exponent. Numerical and other mathematical evidence, called first digit counting compatibility, is provided for this statement.


first digit; Niven number; asymptotic counting function; probabilistic number theory; mean absolute deviation; probability weighted least squares

Full Text:



Benford, F. (1938). The law of anomalous numbers. Proc. Amer. Phil. Soc. 78, 551-572.

Berger, A. and T.P. Hill (2015). An Introduction to Benford’s Law. Princeton University Press, Princeton, New Jersey.

De Koninck, J.M. and N. Doyon (2003). On the number of Niven numbers up to x. Fibonacci Quarterly 41(5), 431-440.

De Koninck, J.M., Doyon N. and I. Kátai (2003). On the counting function for the Niven numbers. Acta Arithmetica 106, 265-275.

Author (2014-2016) 6 papers

Kennedy, R.E. and C.N. Cooper (1984). On the natural density of the Niven numbers. College Math. J. 15, 309-312.

Miller, S.J. (2015). (Editor). Benford’s Law: Theory and Applications. Princeton University Press, Princeton, New Jersey.

Newcomb, S. (1881). Note on the frequency of use of the different digits in natural numbers. Amer. J. Math. 4, 39-40.

Pietronero, L. Tossati, E., Tossati, V. and A. Vespignani (2000). Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf, Physica A 293, 297-304.


  • There are currently no refbacks.

Copyright (c) 2016 Journal of Progressive Research in Mathematics

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Copyright © 2016 Journal of Progressive Research in Mathematics. All rights reserved.

ISSN: 2395-0218.

For any help/support contact us at,