A083379 a(n) = the number of squares with at most n digits and first digit 1.

1, 2, 7, 20, 62, 193, 608, 1918, 6061, 19160, 60582, 191568, 605782, 1915640, 6057776, 19156359, 60577716, 191563545, 605777108, 1915635402

Offset: 1

Author

Werner S. Hürlimann (whurlimann(AT)bluewin.ch), Jun 05 2003

Keywords

• base
• easy
• nonn

Comments

Asymptotically, the probability that a square begins with 1 is (sqrt(2)-1)/(sqrt(10)-1).

A generalization to arbitrary powers is found in Hürlimann, 2004. As the power increases, the probability distribution approaches Benford's law.

Links

• Robert Israel, Table of n, a(n) for n = 1..1999
• W. Hürlimann, Integer powers and Benford's law, International Journal of Pure and Applied Mathematics, vol. 11, no. 1, pp. 39-46, 2004.
• Index entries for sequences related to Benford's law

Crossrefs

Cf. A083377, A083378, A083380.

Programs

Maple

ListTools:-PartialSums([seq(floor(sqrt(2*10^n))-ceil(sqrt(10^n))+1, n=0..20)]); # Robert Israel, Feb 15 2021

Extensions

Edited by Don Reble, Nov 05 2005