A010461 -id:A010461 - cp's OEIS frontend

A122986 Squares mod 1000.

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

A187127 Triangular numbers k*(k+1)/2 mod 100, 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, 53, 55, 56, 60, 61, 65, 66, 70, 71, 75, 76, 78, 80, 81, 85, 86, 90, 91, 95, 96

Offset: 1

Shyam Sunder Gupta, Aug 30 2013

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

All triangular numbers less than 100 belong to this sequence. The sequence is finite with 44 terms: a(44)=96 is the last term.

			The last two digits of k*(k+1)/2  can be 00, 01, 03, 20, 91, etc., but not 02, 04, 12, 97, etc.

Cf. A000217, A010461.

Union[Table[Mod[n*(n + 1)/2, 100], {n, 1, 100}]]

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

A002015 a(n) = n^2 reduced mod 100.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Periodic with period 50: (0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 21, 44, 69, 96, 25, 56, 89, 24, 61, 0, 41, 84, 29, 76, 25, 76, 29, 84, 41, 0, 61, 24, 89, 56, 25, 96, 69, 44, 21, 0, 81, 64, 49, 36, 25, 16, 9, 4, 1) and next term is 0. The period is symmetrical about the "midpoint" 25. - Zak Seidov, Oct 26 2009

A010461 gives the range of this sequence. - Reinhard Zumkeller, Mar 21 2010

Index entries for sequences related to final digits of numbers
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,1).

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

PowerMod[Range[0,60],2,100] (* Harvey P. Dale, Nov 28 2012 *)

a(n)=n^2%100 \\ Charles R Greathouse IV, Oct 07 2015

From Reinhard Zumkeller, Mar 21 2010: (Start)
a(n) = (n mod 10) * ((n mod 10) + 20 * ((n\10) mod 10)) mod 100.
a(n) = A174452(n) mod 100; A008959(n) = a(n) mod 10;
a(m*n) = a(m)*a(n) mod 100;
a(n) = (n mod 100)^2 mod 100;
a(n) = n for n = 0, 1, and 25. (End)

Definition rephrased at the suggestion of Zak Seidov, Oct 26 2009

A028813 Nonsquares mod 100.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 26, 27, 28, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 42, 43, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 82, 83, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 97, 98, 99

Offset: 1

N. J. A. Sloane

Complement of A010461. Numbers n such that k^2 == n (mod 100) has no solution. - Altug Alkan, Dec 10 2015

Cf. A000037 (nonsquares).

Cf. A010461.

n = 100; Complement[Range[0, n - 1], Union@ Array[PowerMod[#, 2, n] &, {100}]] (* Michael De Vlieger, Dec 10 2015 *)

v=vector(100,n,n^2%100); v=vecsort(v); for(i=1, 99, if (!vecsearch(v,i), print1(i,", "))) \\ Derek Orr, Sep 25 2014

for (n=0, 99, if (!issquare(Mod(n, 100)), print1(n, ", "));); \\ Michel Marcus, Dec 10 2015

(0 to 99).diff((1 to 100).map(n => (n * n) % 100)) // Alonso del Arte, Dec 25 2019

More terms to show full sequence from Derek Orr, Sep 25 2014

A210251 Residues modulo 100 of odd squares.

Original entry on oeis.org

1, 9, 21, 25, 29, 41, 49, 61, 69, 81, 89

Offset: 1

M. F. Hasler, Mar 19 2012

Range of A016754. Odd terms from A010461. See also A002015, A008959, A174452.

Also the range of A030156 and A192775 without initial term.

Mod[#,100]&/@(Range[1,55,2]^2)//Union (* Harvey P. Dale, Jul 27 2017 *)

PARI
```
vecsort(vector(12,n,(2*n-1)^2)%100,,8)
```

{1,9} + {0,1,2,3,4}*20 union {25}.

A121996 Sums of two squares mod 100.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 62, 64, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 89, 90, 92, 93, 94, 96, 97, 98

Offset: 1

Zak Seidov, Sep 11 2006

All 72 possible values of ((m^2+n^2) mod 100) for all m's and n's. Not-sums of two squares mod 100 are: 3, 6, 7, 11, 12, 14, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99 (28 terms).

Cf. A010461 Squares mod 100, A001481 Sums of 2 squares.

Cf. A010461, A001481.

A228642 Squares of primes mod 100.

Original entry on oeis.org

1, 4, 9, 21, 25, 29, 41, 49, 61, 69, 81, 89

Offset: 1

Richard R. Forberg, Aug 28 2013

a(n) is the complete list of the last two digits of terms in A001248, with leading zeros omitted.
Numbers 4 and 25 appear only once as the squares of 2 and 5 respectively.
This leaves only 10 different digit pairs that recur in A001248.
Contrast with A010461 which has 22 different values for the squares of the integers (see A000290), all of which recur.
Nearly identical to A210251, for the squares of odd numbers.

Cf. A001248, A000290.

cp's OEIS Frontend

A122986 Squares mod 1000.

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A187127 Triangular numbers k*(k+1)/2 mod 100, sorted and uniqued.

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

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A002015 a(n) = n^2 reduced mod 100.

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A028813 Nonsquares mod 100.

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A210251 Residues modulo 100 of odd squares.

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A121996 Sums of two squares mod 100.

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A228642 Squares of primes mod 100.

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