A036688 Number of distinct n-digit suffixes of base-10 squares not containing the digit 0.
5, 18, 119, 698, 5449, 41735, 359207, 3085197, 27434602, 243921771, 2188569304, 19636586858
Offset: 1
Examples
Any square ends with one of [ 0 ], 1, 4, 5, 6, 9, so a(1) = 5.
a(3) = A000993(3) - a(2) - #{100, 104, 201, 204, 209, 304, 400, 401, 404, 409, 500, 504, 600, 601, 604, 609, 704, 801, 804, 809, 900, 904} = 159 - 18 - 22 = 119, cf. A122986. - _Reinhard Zumkeller_, Mar 21 2010
Links
- Josiah H. Drummond, Problem 57, Amer. Math. Monthly, Vol. 5 (1898), p. 26; reprinted on p. 906 of Vol. 105 (1998).
Crossrefs
Cf. A036788.
Programs
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Mathematica
(* A partly empirical script *) a[n_] := (Clear[qr]; qr[] = False; For[k = 1, k <= 10^n/4, k++, m = PowerMod[k, 2, 10^n]; If[m > 10^(n-1) && FreeQ[IntegerDigits[m], 0], qr[m] = True]]; For[cnt = 0; k = 10^(n-1)+1, k <= 10^n-1, k++, If[qr[k], cnt++]]; cnt); a[1] = 5; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 10}] (* _Jean-François Alcover, Jul 31 2015 *)
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Python
from math import isqrt def a(n): suffixes = set() for k in range(isqrt(10 ** (n - 1)) + 1, 10 ** n): kk = k * k s = str(kk)[-n:] if "0" not in s and len(s) >= n: suffixes.add(s) return len(suffixes) print([a(n) for n in range(1, 8)]) # Michael S. Branicky, May 18 2021
Extensions
Explanation and more terms from David W. Wilson
a(11)-a(12) from Bert Dobbelaere, Mar 10 2021