A033569 -id:A033569 - cp's OEIS frontend

A008959 Final digit of squares: a(n) = n^2 mod 10.

0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0

Offset: 0

N. J. A. Sloane, Mar 15 1996

a(m*n) = a(m)*a(n) mod 10; a(5*n+k) = a(5*n-k) for k <= 5*n. - Reinhard Zumkeller, Apr 24 2009
a(n) = n^6 mod 10. - Zerinvary Lajos, Nov 06 2009
a(n) = A002015(n) mod 10 = A174452(n) mod 10. - Reinhard Zumkeller, Mar 21 2010
Decimal expansion of 166285490/1111111111. - Alexander R. Povolotsky, Mar 09 2013

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for sequences related to final digits of numbers
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Cf. A000290, A070431, A070435, A070438, A070442, A070452, A159852, A010879, A008960, A070514.

[0] cat [Intseq(n^2)[1]: n in [1..80]]; // Bruno Berselli, Feb 14 2013

[n^2 - 10*Floor(n^2/10): n in [0..80]]; // Vincenzo Librandi, Jun 16 2015

A008959:=n->(n^2 mod 10); seq(A008959(n), n=0..50); # Wesley Ivan Hurt, Jun 06 2014

Table[Mod[n^2,10],{n,0,200}] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2011 *)
PowerMod[Range[0,80],2,10] (* or *) LinearRecurrence[{0,0,0,0,0,0,0,0,0,1},{0,1,4,9,6,5,6,9,4,1},120] (* Harvey P. Dale, Oct 16 2012 *)

a(n)=n^2%10 \\ Charles R Greathouse IV, Sep 24 2015

[power_mod(n,2,10) for n in range(0, 81)] # Zerinvary Lajos, Nov 06 2009

Periodic with period 10. - Franklin T. Adams-Watters, Mar 13 2006
a(n) = 4.5 - (1 + 5^(1/2))*cos(Pi*n/5) + (-1 - 3/5*5^(1/2))*cos(2*Pi*n/5) + (5^(1/2) - 1)*cos(3*Pi*n/5) + (-1 + 3/5*5^(1/2))*cos(4*Pi*n/5) - 0.5*(-1)^n. - Richard Choulet, Dec 12 2008
a(n) = A010879(A000290(n)). - Reinhard Zumkeller, Jan 04 2009
G.f.: (x^9+4*x^8+9*x^7+6*x^6+5*x^5+6*x^4+9*x^3+4*x^2+x)/(-x^10+1). - Colin Barker, Aug 14 2012
a(n) = n^2 - 10*floor(n^2/10). - Wesley Ivan Hurt, Jun 12 2013
a(n) = (n - 5*A002266(n + 2))^2 + 5*(5*A002266(n + 2) mod 2). - Wesley Ivan Hurt, Jun 06 2014
a(n) = A033569(n+3) mod 10. - Wesley Ivan Hurt, Dec 06 2014
a(n) = n^k mod 10; for k > 0 where k mod 4 = 2. - Doug Bell, Jun 15 2015

A259758 Squarefree semiprimes of the form (2p - 3) (3*p - 2), p prime.

Original entry on oeis.org

21, 91, 209, 589, 851, 2881, 7739, 10541, 16171, 26069, 29329, 75151, 95129, 110839, 165169, 194219, 216409, 220991, 264389, 374749, 411601, 653069, 717949, 829931, 1108969, 1119311, 1171741, 1269139, 1416689, 2059789, 3161729, 3374249, 3428459, 4924109

Offset: 1

Reinhard Zumkeller, Jul 05 2015

nonn

a(n) = (2*A259730(n) - 3) * (3*A259730(n) - 2);

3431 = A033569(24) = (2*25-3)*(3*25-2) = 47*73 = A006881(946) is the smallest term in the intersection of A006881 and A033569 not belonging to this sequence.

			.    n | p = A259730(n) | 2*p - 3 | 3*p - 2 |   a(n)
.  ----+----------------+---------+---------+--------
.    1 |              3 |       3 |       7 |     21
.    2 |              5 |       7 |      13 |     91
.    3 |              7 |      11 |      19 |    209
.    4 |             11 |      19 |      31 |    589
.    5 |             13 |      23 |      37 |    851
.    6 |             23 |      43 |      67 |   2881
.    7 |             37 |      71 |     109 |   7739
.    8 |             43 |      83 |     127 |  10541
.    9 |             53 |     103 |     157 |  16171
.   10 |             67 |     131 |     199 |  26069
.   11 |             71 |     139 |     211 |  29329
.   12 |            113 |     223 |     337 |  75151  .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A259730, subsequence of A006881, subsequence of A033569.

a259758 n = (2 * p - 3) * (3 * p - 2)  where p = a259730 n

Select[Table[(2p-3)(3p-2),{p,Prime[Range[200]]}],PrimeOmega[#]==2&&SquareFreeQ[ #]&] (* Harvey P. Dale, Jul 20 2022 *)

a(n) = 6*A259730(n)^2 - 13*A259730(n) + 6.

cp's OEIS Frontend

A008959 Final digit of squares: a(n) = n^2 mod 10.

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A259758 Squarefree semiprimes of the form (2p - 3) (3*p - 2), p prime.

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cp's OEIS Frontend

A008959 Final digit of squares: a(n) = n^2 mod 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Maple

Mathematica

PARI

Sage

Formula

A259758 Squarefree semiprimes of the form (2*p - 3) * (3*p - 2), p prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A259758 Squarefree semiprimes of the form (2p - 3) (3*p - 2), p prime.