A089951 Numbers having the same leading decimal digits as their squares.

0, 1, 10, 11, 12, 13, 14, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 895

Offset: 1

Author

Reinhard Zumkeller, Jan 12 2004

Keywords

• nonn
• base

Comments

A000030(a(n)) = A002993(a(n)) = A000030(A000290(a(n))).

Examples

			895*895 = 801025, therefore 895 is a term: a(55)=895.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A018834.

Cf. A144582. - Reinhard Zumkeller, Aug 17 2008

Cf. A000030, A002993, A000290, A256523 (subsequence).

Programs

Haskell

a089951 n = a089951_list !! (n-1)
a089951_list = [x | x <- [0..], a000030 x == a000030 (x ^ 2)]
-- Reinhard Zumkeller, Apr 01 2015

Maple

F:= proc(d) $10^d .. floor(sqrt(2)*10^d), $ ceil(sqrt(80)*10^d) .. 9*10^d - 1, $ ceil(sqrt(90)*10^d) .. 10^(d+1)-1 end proc:
0, F(0), F(1), F(2), F(3); # Robert Israel, Mar 18 2015

Mathematica

d[n_] := IntegerDigits[n]; Select[Range[895],
First[d[#]] == First[d[#^2]] &] (* Jayanta Basu, Jun 03 2013 *)

PARI

a(n)={my(v = [1, sqrt(80), sqrt(90)], w=[(k)->10^k * ((sqrt(2) - 1))\1 + 1, (k)->9 * 10^k - ceil(sqrt(80) * 10^k), (k)->10 * 10^k - ceil(sqrt(90) * 10^k)],i = 1,k = 0); if(n==1, 0, n--; while(n>w[i](k), n-=w[i](k); i++; if(i == 4, i = 1; k++)); ceil(v[i]*10^k)+n-1)} \\ David A. Corneth, Feb 26 2015

PARI

isok(n) = (n == 0) || (digits(n)[1] == digits(n^2)[1]); \\ Michel Marcus, Mar 18 2015

Formula

A number n is in the sequence iff n = 0 or n/10^floor(log_10(n)) lies in one of the half-open intervals [1, sqrt(2)), [sqrt(80), 9) or [sqrt(90), 10). - David W. Wilson, May 29 2008