

A220468


Number of cyclotomic cosets of n mod 10.


4



1, 10, 2, 4, 3, 2, 5, 4, 2, 6, 1, 10, 2, 4, 3, 2, 5, 4, 2, 6, 1, 10, 2, 4, 3, 2, 5, 4, 2, 6, 1, 10, 2, 4, 3, 2, 5, 4, 2, 6, 1, 10, 2, 4, 3, 2, 5, 4, 2, 6, 1, 10, 2, 4, 3, 2, 5, 4, 2, 6, 1, 10, 2, 4, 3, 2, 5, 4, 2, 6, 1, 10, 2, 4, 3, 2, 5, 4, 2, 6, 1, 10, 2, 4, 3, 2, 5, 4, 2, 6
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OFFSET

0,2


COMMENTS

In other words, number of different cycles mod 10 obtained by repeatedly multiplying by n, with different starting elements. There are four different cycles for n = 3: {1, 3, 9, 7}, {2, 6, 8, 4}, {5}, {0}. By starting with a random number below 10, the numbers obtained by repeatedly multiplying by 3 and then taking modulo 10 repeat through one of these four different cycles.
There are two different cycles for n = 2: {2, 4, 8, 6}, {0}. Note that this does not cover all the positive integers less than 10. The elements 1, 3, 7, 9 belong to the cycle {2, 4, 8, 6} since on starting with them, the numbers obtained by the process of repeatedly multiplying by 2 and then taking modulo 10 repeat through the elements of this cycle. Likewise the element 5 belongs to the cycle {0}. For n coprime to 10, the different cycles will cover all the elements less than 10 and will form different equivalence classes by themselves. For other values of n, the cycles will not cover all the elements less than 10.
This sequence is periodic with period of 10, since x^i == (x + 10)^i mod 10.


LINKS

Table of n, a(n) for n=0..89.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).


FORMULA

G.f.: (6*x^9+2*x^8+4*x^7+5*x^6+2*x^5+3*x^4+4*x^3+2*x^2+10*x+1) / (x^101).  Colin Barker, Apr 13 2013


EXAMPLE

The following are the different cycles obtained by repeatedly multiplying by n, and then taking mod 10, with different starting elements.
n = 0: {0}.
n = 1: {0}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}.
n = 2: {2, 4, 8, 6}, {0}.
n = 3: {1, 3, 9, 7}, {2, 6, 8, 4}, {5}, {0}.
n = 4: {4, 6}, {2, 8}, {0}.
n = 5: {5}, {0}.
n = 6: {2}, {4}, {6}, {8}, {0}.
n = 7: {1, 7, 9, 3}, {2, 4, 8, 6}, {5}, {0}.
n = 8: {8, 4, 2, 6}, {0}.
n = 9: {1, 9}, {3, 7}, {2, 8}, {4, 6}, {5}, {0}.


MATHEMATICA

iter[n_] := Table[ FixedPoint[ Union[#, Mod[n*#, 10]] &, {m}], {m, 0, 9}]; classes[n_] := iter[n] //. {a___List, b_List, c___List, d_List, e___List} /; Intersection[b, d] != {} :> {a, Union[b, d], c, e}; a[n_] := Length[classes[n]]; Table[a[n], {n, 0, 89}] (* JeanFrançois Alcover, Jan 08 2013 *)


PROG

(PARI) k=10; j=1; for(i=0, 100, m=0; n=vector(k, X, 1); for(l=0, k1, if(n[((l*i^j)%k)+1]>=0, n[l+1]=n[((l*i^j)%k)+1]; continue, n[l+1]=m; p=l; for(o=1, eulerphi(k), p=(p*i)%k; if(n[p+1]>1, break); n[p+1]=m); m++)); print1(m", "))
(PARI) a(n)=[1, 10, 2, 4, 3, 2, 5, 4, 2, 6][n%10+1] \\ Charles R Greathouse IV, Jan 08 2013


CROSSREFS

Cf. A173635, A220022A220026.
Sequence in context: A160136 A322467 A049296 * A169851 A161995 A069036
Adjacent sequences: A220465 A220466 A220467 * A220469 A220470 A220471


KEYWORD

nonn,base,less,easy


AUTHOR

V. Raman, Jan 03 2013


STATUS

approved



