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

1,3

Comments

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

Links

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

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A018834.

Cf. A144582. - Reinhard Zumkeller, Aug 17 2008

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

Sequence in context: A228774 A073527 A008707 * A058945 A270040 A339093

Adjacent sequences: A089948 A089949 A089950 * A089952 A089953 A089954

Keyword

nonn,base

Author

Reinhard Zumkeller, Jan 12 2004

Status

approved