login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A323019 a(n) is the smallest k such that A316506(k) = n. 0
1, 2, 4, 8, 20, 40, 120, 520, 1560, 8840, 26520, 185640, 769080, 5383560, 28455960, 199191720, 1166694360, 8166860520, 61834801080, 432843607560, 3771922865880, 26403460061160, 275350369209240, 1927452584464680, 21201978429111480, 171543280017356520 (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 Gaussian integers modulo k is n.

LINKS

Table of n, a(n) for n=0..25.

FORMULA

a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 8. Let p(n) be the n-th prime congruent to 1 modulo 4, q(n) be the n-th prime congruent to 3 modulo 4. Then there exists {i(n)} and {j(n)} such that i(2) = j(2) = i(3) = j(3) = 0; for n >= 4, 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(2) = 4, i(2) = 0, j(2) = 0;

a(3) = 8, i(3) = 0, j(3) = 0;

For n = 4, a(n-2)*p(i(n-2)+1) = a(2)*p(1) = 4*5 = 20, a(n-1)*q(j(n-1)+1) = a(3)*q(1) = 8*3 = 24. So a(4) = 20, i(4) = i(2) + 1 = 1, j(4) = j(2) = 0.

For n = 5, a(n-2)*p(i(n-2)+1) = a(3)*p(1) = 8*5 = 40, a(n-1)*q(j(n-1)+1) = a(4)*q(1) = 20*3 = 60. So a(5) = 40, 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(2) = 20*13 = 260, a(n-1)*q(j(n-1)+1) = a(5)*q(1) = 40*3 = 120. So a(6) = 120, i(6) = i(5) = 1, j(6) = j(5) + 1 = 1.

...

List of the multiplicative groups of Gaussian integers modulo members of this sequence:

a(0) = 1: the trivial group;

a(1) = 2: C_2;

a(2) = 4: C_2 X C_4;

a(3) = 8: C_2 X C_4 X C_4;

a(4) = 20: C_2 X C_4 X C_4 X C_4;

a(5) = 40: C_2 X C_4 X C_4 X C_4 X C_4;

a(6) = 120: C_2 X C_4 X C_4 X C_4 X C_4 X C_8;

a(7) = 520: C_2 X C_4 X C_4 X C_4 X C_4 X C_12 X C_12;

a(8) = 1560: C_2 X C_4 X C_4 X C_4 X C_4 X C_4 X C_12 X C_24;

a(9) = 8840: C_2 X C_4 X C_4 X C_4 X C_4 X C_4 X C_4 X C_48 X C_48;

a(10) = 26520: C_2 X C_4 X C_4 X C_4 X C_4 X C_4 X C_4 X C_8 X C_48 X C_48;

...

PROG

(PARI)

p(n) = my(i=0, k=0); while(i<n, k++; if(prime(k)%4==1, i++)); prime(k)

q(n) = my(i=0, k=0); while(i<n, k++; if(prime(k)%4==3, i++)); prime(k)

a(n) = if(n<3, 2^n, my(v=vector(n), s=vector(n), t=vector(n)); [v[2], v[3]]=[4, 8]; for(i=4, 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

Cf. A316506.

Sequence in context: A283047 A236397 A272122 * A105319 A051389 A078006

Adjacent sequences:  A323016 A323017 A323018 * A323020 A323021 A323022

KEYWORD

nonn

AUTHOR

Jianing Song, Jan 10 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 15 18:26 EDT 2019. Contains 328037 sequences. (Running on oeis4.)