The OEIS is supported by the many generous donors to the OEIS Foundation. Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A307371 Numbers k such that the digits of sqrt(k) begin with k. 9
 0, 1, 98, 99, 100, 9998, 9999, 10000, 999998, 999999, 1000000, 99999998, 99999999, 100000000, 9999999998, 9999999999, 10000000000, 999999999998, 999999999999, 1000000000000, 99999999999998, 99999999999999, 100000000000000, 9999999999999998, 9999999999999999 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS From Chai Wah Wu, Jan 17 2020: (Start) Theorem: A number n is a term if and only if n is 0, 1, 10^(2m), 10^{2m)-1 or 10^{2m}-2 for some m >= 1. Proof: k <= sqrt(k)*10^m < k+1. For m = 0, the only solutions are 0 and 1. For m > 0, k^2 <= k*10^(2m) < (k+1)^2. This is equivalent to k <= 10^2m < k + 2 + 1/k, i.e., 10^(2m)-2-1/k < k <= 10^(2m). Thus the only solutions for k are 10^(2m), 10^(2m)-1 and 10^(2m)-2. (End) LINKS Table of n, a(n) for n=1..25. Dmitry Kamenetsky, Java program to compute terms FORMULA From Chai Wah Wu, Jan 17 2020: (Start) a(n) = 101*a(n-3) - 100*a(n-6) for n > 6. G.f.: x^2*(100*x^4 - x^3 + 99*x^2 + 98*x + 1)/(100*x^6 - 101*x^3 + 1). (End) EXAMPLE sqrt(9998) = 99.989..., which begins with "9998", so 9998 is in the sequence. PROG (Python) A307371_list = [0, 1, 98, 99, 100, 9998] for _ in range(100): A307371_list.append(101*A307371_list[-3]-100*A307371_list[-6]) # Chai Wah Wu, Jan 18 2020 CROSSREFS Cf. A307588, A307600. Sequence in context: A120310 A129887 A205067 * A273460 A095605 A095589 Adjacent sequences: A307368 A307369 A307370 * A307372 A307373 A307374 KEYWORD nonn,base AUTHOR Dmitry Kamenetsky, Apr 17 2019 EXTENSIONS a(12)-a(25) from Jon E. Schoenfield, May 01 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 2 23:17 EST 2023. Contains 367526 sequences. (Running on oeis4.)