|
| |
|
|
A323020
|
|
a(n) is the smallest k such that A319447(k) = n.
|
|
0
|
|
|
|
1, 2, 4, 8, 18, 56, 126, 630, 1638, 8190, 31122, 155610, 964782, 4823910, 35696934, 178484670, 1534968162, 7674840810, 84423248910, 468165289410, 5149818183510, 31367074390470, 345037818295170, 2289796430504310, 25187760735547410, 180893918009840490
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
a(n) is the smallest k such that the rank of the multiplicative group of Eisenstein integers modulo k is n.
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 8, a(4) = 18. Let p(n) be the n-th prime congruent to 1 modulo 6, q(n) be the n-th prime congruent to 5 modulo 6. Then there exists {i(n)} and {j(n)} such that i(3) = j(3) = i(4) = j(4) = 0; for n >= 5, if a(n-2)*p(i(n-2)+1) < a(n-1)*q(j(n-1)+1), then a(n) = a(n-2)*p(i(n-2)+1), i(n) = i(n-2) + 1, j(n) = j(n-2), or a(n) = a(n-1)*q(j(n-1)+1), i(n) = i(n-1), j(n) = j(n-1) + 1.
|
|
|
EXAMPLE
|
a(3) = 8, i(3) = 0, j(3) = 0;
a(4) = 18, i(4) = 0, j(4) = 0;
For n = 5, a(n-2)*p(i(n-2)+1) = a(3)*p(1) = 8*7 = 56, a(n-1)*q(j(n-1)+1) = a(4)*q(1) = 18*5 = 90. So a(5) = 56, i(5) = i(3) + 1 = 1, j(5) = j(3) = 0.
For n = 6, a(n-2)*p(i(n-2)+1) = a(4)*p(1) = 18*7 = 126, a(n-1)*q(j(n-1)+1) = a(5)*q(1) = 56*5 = 280. So a(6) = 126, i(6) = i(4) + 1 = 2, j(6) = j(4) = 0.
For n = 7, a(n-2)*p(i(n-2)+1) = a(5)*p(2) = 56*13 = 728, a(n-1)*q(j(n-1)+1) = a(6)*q(1) = 126*5 = 630. So a(7) = 630, i(7) = i(6) = 1, j(7) = j(6) + 1 = 1.
...
List of the multiplicative groups of Eisenstein integers modulo members of this sequence:
a(0) = 1: the trivial group;
a(1) = 2: C_3;
a(2) = 4: C_2 X C_6;
a(3) = 8: C_2 X C_2 X C_12;
a(4) = 18: C_3 X C_3 X C_3 X C_6;
a(5) = 56: C_2 X C_2 X C_6 X C_6 X C_12;
a(6) = 126: C_3 X C_3 X C_3 X C_6 X C_6 X C_6;
a(7) = 630: C_3 X C_3 X C_3 X C_6 X C_6 X C_6 X C_24;
a(8) = 1638: C_3 X C_3 X C_3 X C_6 X C_6 X C_6 X C_12 X C_12;
a(9) = 8190: C_3 X C_3 X C_3 X C_6 X C_6 X C_6 X C_6 X C_6 X C_24;
a(10) = 31122: C_3 X C_3 X C_3 X C_6 X C_6 X C_6 X C_6 X C_6 X C_36 X C_36;
...
|
|
|
PROG
|
(PARI)
p(n) = my(i=0, k=0); while(i<n, k++; if(prime(k)%6==1, i++)); prime(k)
q(n) = my(i=0, k=0); while(i<n, k++; if(prime(k)%6==5, i++)); prime(k)
a(n) = if(n<4, 2^n, my(v=vector(n), s=vector(n), t=vector(n)); [v[3], v[4]]=[8, 18]; for(i=5, n, my(a=v[i-2]*p(s[i-2]+1), b=v[i-1]*q(t[i-1]+1)); if(a<b, [v[i], s[i], t[i]] = [a, s[i-2]+1, t[i-2]], [v[i], s[i], t[i]] = [b, s[i-1], t[i-1]+1])); v[n])
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
