A122986 -id:A122986 - cp's OEIS frontend

A010461 Squares mod 100.

0, 1, 4, 9, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, 96

Offset: 1

Range of A002015; subset of A122986. - Reinhard Zumkeller, Mar 21 2010

Cf. A010382, A010411, A010462, A010421. - Reinhard Zumkeller, Mar 21 2010
Row 100 of A096008.
Cf. A028813.

[n: n in [0..99] | IsSquare(R! n) where R:= ResidueClassRing(100)]; // Vincenzo Librandi, Dec 28 2019

Union[PowerMod[Range[100], 2, 100]] (* Alonso del Arte, Dec 25 2019 *)

A010461=Set(vector(100,n,n^2)%100) \\ M. F. Hasler, Mar 03 2014

[quadratic_residues(100)] # Zerinvary Lajos, May 28 2009

(1 to 100).map(n => (n * n) % 100).toSet.toSeq.sorted // Alonso del Arte, Dec 25 2019

A000993 Number of distinct quadratic residues mod 10^n; also number of distinct n-digit endings of base-10 squares.

Original entry on oeis.org

1, 6, 22, 159, 1044, 9121, 78132, 748719, 7161484, 70800861, 699869892, 6978353179, 69580078524, 695292156201, 6947835288052, 69465637212039, 694529215501164, 6944974263529141, 69446563720728612, 694457689921141299, 6944497426351013404

Offset: 0

N. J. A. Sloane

			Any square ends with one of 0, 1, 4, 5, 6, 9, so a(1) = 6.
A square may end with 22 different two-digit combinations: 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, 96. E.g., no number ending with 14 can be square, etc. See also A075821, A075823.
The finite sequence A122986 has a(3) = 159 terms. - _Reinhard Zumkeller_, Mar 21 2010

Albert H. Beiler, Recreations in the Theory of Numbers, Dover Publ., 2nd Ed., NY, 1966, Chapter XV, 'On The Square', p. 139.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Shyam Sunder Gupta, Elegance of Squares, Cubes, and Higher Powers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 2, 29-81.
Walter Penney, On the final digits of squares, Amer. Math. Monthly, 67 (1960), 1000-1002.
Index entries for sequences related to final digits of numbers
Index entries for linear recurrences with constant coefficients, signature (10,30,-300,-129,1290,100,-1000).

Cf. A036688, A023105, A039300-A039306, A075821, A075823.

[1] cat [(83 + 27*(-1)^n + 9*2^(1 + n) + (-1)^n*2^(2 + n) + 9*5^(2 + n) + (-1)^n*5^(2 + n) + 2^(1 + n)*5^(2 + n))/ 72: n in [0..20]]; // Vincenzo Librandi, Mar 29 2012

-(-6+38*z+241*z^2-594*z^3-1285*z^4+1600*z^5+1500*z^6)/((-1+z)*(5*z-1)*(2*z+1)*(2*z-1)*(5*z+1)*(10*z-1)*(z+1)); #  Bruno Salvy

a[n_] := (83 - 27*(-1)^n + 9*2^(n) - (-1)^n*2^(1 + n) + 9*5^(1 + n) - (-1)^n*5^(1 + n) + 2^(n)*5^(1 + n))/72; Table[ Floor[ a[n]], {n, 0, 20}]
(* Or *) a[0] = 1; a[1] = 6; a[2] = 22; a[3] = 159; a[4] = 1044; a[5] = 9121; a[6] = 78132; a[7] = 748719; a[8] = 7161484; a[n_] := 130 a[n - 2] - 3129 a[n - 4] + 13000 a[n - 6] - 10000 a[n - 8]; Table[ a[n], {n, 0, 20}]
(* Or *) CoefficientList[ Series[(1 - 4*x - 68*x^2 + 59*x^3 + 723*x^4 - 5*x^5 - 1700*x^6 - 500*x^7)/(1 - 10*x - 30*x^2 + 300*x^3 + 129*x^4 - 1290*x^5 - 100*x^6 + 1000*x^7), {x, 0, 20}], x] (* Robert G. Wilson v, Nov 27 2004 *)
LinearRecurrence[{10,30,-300,-129,1290,100,-1000},{1,6,22,159,1044,9121,78132,748719},20] (* Harvey P. Dale, Dec 17 2017 *)

print([(2 + 2**n // 6) * (1 + 5**(n+1) // 12) if n else 1 for n in range(21)]) # Nick Hobson, Mar 10 2024

a(n) = floor( (83 - (-1)^n*(27 + 2^(n+1) + 5^(n+1)) + 9*2^n + (9 + 2^n)*5^(n+1)) / 72 ).
a(n+8) = 130 a(n+6) - 3129 a(n+4) + 13000 a(n+2) - 10000 a(n) for n >= 1.
G.f.: (1 - 4*x - 68*x^2 + 59*x^3 + 723*x^4 - 5*x^5 - 1700*x^6 - 500*x^7)/(1 - 10*x - 30*x^2 + 300*x^3 + 129*x^4 - 1290*x^5 - 100*x^6 + 1000*x^7).

A174452 a(n) = n^2 mod 1000.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 24, 89, 156, 225, 296, 369, 444, 521, 600, 681, 764, 849, 936, 25, 116, 209, 304, 401, 500, 601, 704, 809, 916, 25

Offset: 0

Reinhard Zumkeller, Mar 21 2010

a(n) = A000290(n) for n < 32, but a(32) = 24;
A008959(n) = a(n) mod 10; A002015(n) = a(n) mod 100;
periodic with period 500: a(n+500)=a(n) and a(250*n+k)=a(250*n-k) for k <= 250*n;
a(n) = (n mod 1000)^2 mod 1000;
a(m*n) = a(m)*a(n) mod 1000;
A122986 gives the range of this sequence;
a(n) = n for n = 0, 1, and 376.

			Some calculations for n=982451653, to be realized by hand:
a(n) = (53^2 + 200*6*3) mod 1000 = 6409 mod 1000 = 409;
a(n) = (653^2) mod 1000 = 426409 mod 1000 = 409;
a(n) = a(n mod 500) = a(153) = 409;
a(n) = 965211250482432409 mod 1000 = 409.

Cf. A053879, A070430, A070431, A070432, A070433, A070434, A070435, A070438, A070442, A070452, A159852.

a174452 = (`mod` 1000) . (^ 2)  -- Reinhard Zumkeller, Jul 06 2011

seq(n^2 mod 1000, n=0..55); # Nathaniel Johnston, Jun 22 2011

PowerMod[Range[0,60],2,1000] (* Harvey P. Dale, Feb 08 2022 *)

a(n)=n^2%1000 \\ Charles R Greathouse IV, Apr 06 2016

a(n) = ((n mod 100)^2 + 200 * (floor(n/100) mod 10) * (n mod 10)) mod 1000.

A238712 Numbers in which squares may end (in base 10).

Original entry on oeis.org

0, 1, 4, 5, 6, 9, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 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

Offset: 1

M. F. Hasler, Mar 03 2014

The union of "squares mod 10" (= the first 6 terms) and "squares mod 100" (A010461) and "squares mod 1000" (A122986) etc.

The number of terms < 10^k beginning with k=0: 1, 6, 24, 165, 1101, 9306, 79620, 753462, 7198791, 70919559, ... - Robert G. Wilson v, Sep 04 2014

			6 is in the sequence because 4^2 = 16 ends in the digit 6.
7 is not in the sequence because no square can end with the digit 7.

Derek Orr, Table of n, a(n) for n = 1..9306

Cf. A161355, A246422, A246448 (Complement).

mx = 3; t = Union@ Table[ Mod[n^2, 10^mx], {n, 10^mx/2}]; t = Union@ Flatten@ Table[ Mod[t, 10^m], {m, mx}] (* Robert G. Wilson v, Sep 04 2014 *)

a=[];for(m=1,3,a=setunion(a,Set(vector(10^m,n,n^2)%10^m)));a

If n is present so is n^2. - Robert G. Wilson v, Sep 04 2014

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

A122987 Possible last three digits of n^3 (leading zeros omitted).

Original entry on oeis.org

0, 1, 3, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 24, 27, 29, 31, 32, 33, 37, 39, 41, 43, 47, 48, 49, 51, 53, 56, 57, 59, 61, 63, 64, 67, 69, 71, 72, 73, 77, 79, 81, 83, 87, 88, 89, 91, 93, 96, 97, 99, 101, 103, 104, 107, 109, 111, 112, 113, 117, 119

Offset: 1

Sergio Pimentel, Sep 22 2006

This is a finite sequence. a(505) = 999 is the last term.

			The last three digits of n^3 can be 111, 112, 113, etc. but not 114, 115, 116, etc.

Nathaniel Johnston, Table of n, a(n) for n = 1..505 (full sequence)
Eric Weisstein's World of Mathematics, Cubic Numbers.

Cf. A000578, A122986, A122988.

s:={}: for n from 0 to 999 do s:=s union {n^3 mod 1000}: od: op(s); # Nathaniel Johnston, Jun 22 2011

Union[Mod[Range[1000]^3,1000]] (* James C. McMahon, Nov 14 2024 *)

A122988 Number of possible arrangements of the last three digits of x^n for all x>0 (leading zeros omitted).

Original entry on oeis.org

1, 1000, 159, 505, 52, 105, 102, 505, 52, 505, 22, 505, 52, 505, 102, 105, 52, 505, 102, 505, 12, 505, 102, 505, 52, 25, 102, 505, 52, 505, 22, 505, 52, 505, 102, 105, 52, 505, 102, 505, 12, 505, 102, 505, 52, 105, 102, 505, 52, 505, 6, 505, 52, 505, 102, 105, 52

Offset: 0

Sergio Pimentel, Sep 22 2006

Only possible values are {1, 4, 6, 12, 22, 25, 52, 102, 105, 159, 505, 1000}. - Robert G. Wilson v, Sep 27 2006.

			a(0) = 1 because the last three digits of x^0 are always 001 (just one possibility).
a(100)=4 because the last three digits of x^100 can be 000, 001, 376 or 625 (which is four possibilities).

Robert G. Wilson v, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Cubic Number.
Eric Weisstein's World of Mathematics, Square Number.

Cf. A122986, A122987, A000578, A006716, A027676.

f[n_] := Length@ Union@ PowerMod[ Range@1000, n, 1000]; Table[ f@n, {n, 0, 56}] (* Robert G. Wilson v *)

a(n)=1 for n=0 only,
a(n)=4 for n=100*k, k>=1,
a(n)=6 for n=100*k-50, k>=1,
a(n)=12 for n=20*k, k>=1 except if k == 0 (mod 5),
a(n)=22 for n=20*k-10, k>=1 except if k == 3 (mod 5),
a(n)=25 for n=50*k-25, k>=1,
a(n)=52 for n=4*k, k>=1 except if k == 0 (mod 5),
a(n)=102 for n=4*k-2, k>=2 except if k == 3 (mod 5),
a(n)=105 for n=10*k-5, k>=1 except if k == 3 (mod 5),
a(n)=159 for n=2 only,
a(n)=505 for n=2*k-1, k>=2 except if k == 3 (mod 5) and
a(n)=1000 for n=1 only.

Edited and extended by Robert G. Wilson v, Sep 27 2006

A187126 Triangular numbers k*(k+1)/2 mod 1000, sorted and uniqued.

Original entry on oeis.org

0, 1, 3, 5, 6, 10, 11, 15, 16, 20, 21, 25, 26, 28, 30, 31, 35, 36, 40, 41, 45, 46, 50, 51, 55, 56, 60, 61, 65, 66, 70, 71, 75, 76, 78, 80, 81, 85, 86, 90, 91, 95, 96, 100, 101, 105, 106, 110, 111, 115, 116, 120, 121, 125, 126, 128, 130, 131, 135, 136, 140

Offset: 1

Shyam Sunder Gupta, Aug 30 2013

Possible last three digits of triangular numbers k*(k+1)/2 (leading zeros omitted).

All triangular numbers less than 1000 belong to this sequence. The sequence is finite with 424 terms: a(424)=996 is the last term.

			The last three digits of k*(k+1)/2 can be 000, 001, 003, 021, 140, etc., but not 002, 004, 012, 139 etc.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..424

Cf. A000217, A122986.

Union[Table[Mod[n*(n + 1)/2, 1000], {n, 1, 1000}]]

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A010461 Squares mod 100.

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A000993 Number of distinct quadratic residues mod 10^n; also number of distinct n-digit endings of base-10 squares.

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A174452 a(n) = n^2 mod 1000.

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A238712 Numbers in which squares may end (in base 10).

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

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A122987 Possible last three digits of n^3 (leading zeros omitted).

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A122988 Number of possible arrangements of the last three digits of x^n for all x>0 (leading zeros omitted).

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