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A252911
Irregular triangular array read by rows: T(n,k) is the number of elements in the multiplicative group of integers modulo n that have order k, n>=1, 1<=k<=A002322(n).
2
1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 0, 0, 2, 1, 3, 1, 1, 2, 0, 0, 2, 1, 1, 0, 2, 1, 1, 0, 0, 4, 0, 0, 0, 0, 4, 1, 3, 1, 1, 2, 2, 0, 2, 0, 0, 0, 0, 0, 4, 1, 1, 2, 0, 0, 2, 1, 3, 0, 4, 1, 3, 0, 4, 1, 1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 8, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, 3, 0, 4
OFFSET
1,10
COMMENTS
Row sums are A000010.
Column 2 = A155828(n) = A060594(n) - 1.
LINKS
Eric Weisstein's World of Mathematics, Modulo Multiplication Group.
EXAMPLE
1;
1;
1, 1;
1, 1;
1, 1, 0, 2;
1, 1;
1, 1, 2, 0, 0, 2;
1, 3;
1, 1, 2, 0, 0, 2;
1, 1, 0, 2;
1, 1, 0, 0, 4, 0, 0, 0, 0, 4;
1, 3;
1, 1, 2, 2, 0, 2, 0, 0, 0, 0, 0, 4;
1, 1, 2, 0, 0, 2;
1, 3, 0, 4;
T(15,2)=3 because the elements 4, 11, and 14 have order 2 in the modulo multiplication group (Z/15Z)*. We observe that 4^2, 11^2, and 14^2 are congruent to 1 mod 15.
MAPLE
with(numtheory):
T:= n-> `if`(n=1, 1, (p-> seq(coeff(p, x, j), j=1..degree(p)))(
add(`if`(igcd(n, i)>1, 0, x^order(i, n)), i=1..n-1))):
seq(T(n), n=1..30); # Alois P. Heinz, Dec 30 2014
MATHEMATICA
Table[Table[
Count[Table[
MultiplicativeOrder[a, n], {a,
Select[Range[n], GCD[#, n] == 1 &]}], k], {k, 1,
CarmichaelLambda[n]}], {n, 1, 20}] // Grid
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Dec 24 2014
STATUS
approved