A000993 -id:A000993 - cp's OEIS frontend

A075759 Duplicate of A000993.

1, 6, 22, 159, 1044, 9121, 78132, 748719, 7161484, 70800861, 699869892, 6978353179

Offset: 0

dead

A030466 Squares that are concatenations of two consecutive nonzero numbers.

183184, 328329, 528529, 715716, 60996100, 1322413225, 4049540496, 106755106756, 453288453289, 20661152066116, 29752082975209, 2214532822145329, 2802768328027684, 110213248110213249, 110667555110667556, 147928995147928996, 178838403178838404, 226123528226123529

Offset: 1

Patrick De Geest

British Mathematical Olympiad, 1993, Round 1, Question 1: "Find, showing your method, a six-digit integer n with the following properties: (i) n is a perfect square, (ii) the number formed by the last three digits of n is exactly one greater than the number formed by the first three digits of n. (Thus n might look like 123124, although this is not a square.)"
Steve Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 1 of the British Mathematical Olympiad 1993, page 164.

Iain Fox, Table of n, a(n) for n = 1..10471.
British Mathematical Olympiad 1993, Round 1, Problem 1.
Michael Penn, British Mathematics Olympiad 1993 Round 1 Question 1, YouTube, Apr 24, 2020.
Pante Stanica, Squares as concatenation of consecutive integers, Slides, West Coast Number Theory, Dec 17 2017.
Index to sequences related to Olympiads.

Cf. A000993, A030465, A030467, A054214, A054215, A054216, A020339, A020340.

fQ[n_] := IntegerQ[Sqrt[n*10^Floor[1 + Log10[n + 1]] + n + 1]]; (* Robert G. Wilson v, Dec 27 2017 *)

lista(nn) = forstep(n=183, nn, [3, 5, 7, 5, 3, 1, 4, 7, 5, 3, 5, 7, 5, 3, 5, 7, 5, 3, 5, 7, 4, 1], my(s = eval(concat(Str(n), Str(n+1)))); if(issquare(s), print1(s, ", "))) \\ Iain Fox, Dec 27 2017

eea(x, y) = my(a=max(x,y), b=min(x,y), s=0, so=1, st, r=b, ro=a, rt, q, t); while(r, q=ro\r; rt=r; r=ro-q*r; ro=rt; st=s; s=so-q*s; so=st); t=(ro-so*a)\b; if(x>y, [so, t], [t, so]) \\ Extended Euclidean Algorithm
lista(nn) = my(res=Set(), b, f2, c, s); for(d=3, nn, b=10^d+1; fordiv(b, f, if(f!=1 && f!=b, f2=b/f; if(gcd(f, f2)==1, c=eea(f, f2); if(c[1]<0, s=f*(f2+2*c[1])*f2*(f-2*c[2])+1, s=f*(2*c[1])*f2*(-2*c[2])+1); if(#digits(s)==d*2, res=setunion(res, Set(s))))))); Vec(res) \\ (Will find all values of length nn*2 or shorter) Iain Fox, Oct 16 2021

a(n) = A030465(n)*(10^A055642(A030465(n))+1)+1. - Iain Fox, Oct 16 2021

a(15)-a(17) from Arkadiusz Wesolowski, Apr 02 2014

a(18) from Iain Fox, Dec 27 2017

A122986 Squares mod 1000.

Original entry on oeis.org

0, 1, 4, 9, 16, 24, 25, 36, 41, 44, 49, 56, 64, 76, 81, 84, 89, 96, 100, 104, 116, 121, 124, 129, 136, 144, 156, 161, 164, 169, 176, 184, 196, 201, 204, 209, 216, 224, 225, 236, 241, 244, 249, 256, 264, 276, 281, 284, 289, 296, 304, 316, 321, 324, 329, 336, 344

Offset: 1

Sergio Pimentel, Sep 22 2006

Possible last three digits of n^2 (leading zeros omitted).

Range of A174452; A010461 is a subset; and also all squares less than 1000 belong to this sequence; the sequence is finite with A000993(3)=159 terms: a(159)=996 is the last term.

			The last three digits of n^2 can be 000, 001, 236, 241, 996, etc. but not 002, 003, 237, 238, etc.

R. Zumkeller, Table of n, a(n) for n = 1..159 (full sequence)
Eric Weisstein's World of Mathematics, Square Numbers.
Index entries for sequences related to final digits of numbers

Cf. A036688, A010382, A010411, A010462, A010421, A174452.
Cf. A006716, A053975, A027676, A027678, A122987, A122988.
Row 1000 of A096008.

[n: n in [0..999] | IsSquare(R! n) where R:= ResidueClassRing(1000)]; // Vincenzo Librandi, Dec 29 2019

s:={}: for n from 0 to 999 do s:=s union {n^2 mod 1000}: od: op(s); # Nathaniel Johnston, Jun 22 2011

Union[PowerMod[Range[1000], 2, 1000]] (* Vincenzo Librandi, Dec 29 2019 *)

More terms and additional comments from Reinhard Zumkeller, Mar 21 2010

Edited by N. J. A. Sloane, Apr 10 2010

A036688 Number of distinct n-digit suffixes of base-10 squares not containing the digit 0.

Original entry on oeis.org

5, 18, 119, 698, 5449, 41735, 359207, 3085197, 27434602, 243921771, 2188569304, 19636586858

Offset: 1

R. K. Guy

			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

Josiah H. Drummond, Problem 57, Amer. Math. Monthly, Vol. 5 (1898), p. 26; reprinted on p. 906 of Vol. 105 (1998).

Cf. A036788.

(* 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 *)

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

Explanation and more terms from David W. Wilson

a(11)-a(12) from Bert Dobbelaere, Mar 10 2021

A039306 Number of distinct quadratic residues mod 9^n.

Original entry on oeis.org

1, 4, 31, 274, 2461, 22144, 199291, 1793614, 16142521, 145282684, 1307544151, 11767897354, 105911076181, 953199685624, 8578797170611, 77209174535494, 694882570819441, 6253943137374964, 56285488236374671, 506569394127372034

Offset: 0

David W. Wilson

Number of distinct n-digit suffixes of base 9 squares.
From Danny Rorabaugh, Dec 15 2015: (Start)
Construct the word y_n as follows: y_0 = a; y_{n+1} is three concatenated copies of y_n, except that the middle copy is written with letters not used in y_n. For example:
y_0 = a;
y_1 = aba;
y_2 = abacdcaba;
y_3 = abacdcabaefeghgefeabacdcaba.
a(n) is the number of nonempty subwords of y_n that occur as a subword exactly once.
Let s(n, k) be the number of subwords of y_n that occur exactly 2^k times. One can show that s(n, 0) = a(n) using s(n+1, k+1) = s(n, k) + s(n, k+1), binomial(3^n+1, 2) = Sum_{k=0..n) s(n, k)*2^k, and the formulas for a(n) below.
(End)

			From _Danny Rorabaugh_, Dec 15 2015: (Start)
The squares of the numbers 0..8 are [0, 1, 4, 9, 16, 25, 36, 49, 64]. Modulo 9, these are [0, 1, 4, 0, 7, 7, 0, 4, 1]. Thus there are a(1) = 4 distinct quadratic residues module 9^1 = 9: 0, 1, 4, and 7.
There are a(2) = 31 subwords of y_2 = abacdcaba which occur in y_2 exactly once: [abac, abacd, abacdc, abacdca, abacdcab, abacdcaba, bac, bacd, bacdc, bacdca, bacdcab, bacdcaba, ac, acd, acdc, acdca, acdcab, acdcaba, cd, cdc, cdca, cdcab, cdcaba, d, dc, dca, dcab, dcaba, ca, cab, caba].
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Quadratic residues modulo k^n: A023105 (k=2), A039300 (k=3), A039301 (k=4), A039302 (k=5), A039303 (k=6), A039304 (k=7), A039305 (k=8), this sequence (k=9), A000993 (k=10).

I:=[1, 4, 31]; [n le 3 select I[n] else 9*Self(n-1)+Self(n-2)-9*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Apr 22 2012

CoefficientList[Series[(1-6*x)/((1-x)*(1-9*x)),{x,0,30}],x] (* Vincenzo Librandi, Apr 22 2012 *)

a(n) = floor((9^n+3)*3/8).
G.f.: (1-6*x)/((1-x)*(1-9*x)). - _Colin Barker, Mar 14 2012
a(n) = 9*a(n-1) +a(n-2) -9*a(n-3). - Vincenzo Librandi, Apr 22 2012
a(n) = (5+3^(2n+1))/8 = a(n-1) + 3^(2n-1). - Danny Rorabaugh, Dec 15 2015

A279085 Number of distinct residues of triangular numbers mod 10^n.

Original entry on oeis.org

1, 6, 44, 424, 4176, 41696, 416704, 4166784, 41666816

Offset: 0

Jon E. Schoenfield, Jan 14 2017

Number of distinct n-digit endings of triangular numbers A000217 (written in base 10).

			The last digit of a triangular number is one of 0, 1, 3, 5, 6, or 8, so a(1) = 6. (To verify that no number from {2, 4, 7, 9} can be the last digit of a triangular number T, note that 8*T+1, which must be a square, would end with 7, 3, 7, or 3, respectively, but no square ends with 3 or 7.)
The 44 two-digit combinations with which a triangular number may end are 00, 01, 03, 05, 06, 10, 11, 15, 16, 20, 21, 25, 26, 28, 30, 31, 35, 36, 40, 41, 45, 46, 50, 51, 53, 55, 56, 60, 61, 65, 66, 70, 71, 75, 76, 78, 80, 81, 85, 86, 90, 91, 95, 96. Of these, there are ten combinations of each of the forms x0, x1, x5, and x6; the other four, which are the only ones that end with a digit other than 0, 1, 5, or 6, are 03, 28, 53, and 78 (i.e., numbers whose residue modulo 25 is 3):
.
                            last digit
                   0  1  2  3  4  5  6  7  8  9
                +------------------------------
              0 | 00 01    03    05 06
              1 | 10 11          15 16
              2 | 20 21          25 26    28
next-to-last  3 | 30 31          35 36
    digit     4 | 40 41          45 46
              5 | 50 51    53    55 56
              6 | 60 61          65 66
              7 | 70 71          75 76    78
              8 | 80 81          85 86
              9 | 90 91          95 96

Cf. A000217, A000993.

(Empirical) a(n) = (5*10^n + (9 - 2*(-1)^n)*2^n)/12.

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A075759 Duplicate of A000993.

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A030466 Squares that are concatenations of two consecutive nonzero numbers.

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A122986 Squares mod 1000.

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A036688 Number of distinct n-digit suffixes of base-10 squares not containing the digit 0.

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A039306 Number of distinct quadratic residues mod 9^n.

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A279085 Number of distinct residues of triangular numbers mod 10^n.

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