# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a323019 Showing 1-1 of 1 %I A323019 #9 Jan 23 2019 09:33:14 %S A323019 1,2,4,8,20,40,120,520,1560,8840,26520,185640,769080,5383560,28455960, %T A323019 199191720,1166694360,8166860520,61834801080,432843607560, %U A323019 3771922865880,26403460061160,275350369209240,1927452584464680,21201978429111480,171543280017356520 %N A323019 a(n) is the smallest k such that A316506(k) = n. %C A323019 a(n) is the smallest k such that the rank of the multiplicative group of Gaussian integers modulo k is n. %F A323019 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. %e A323019 a(2) = 4, i(2) = 0, j(2) = 0; %e A323019 a(3) = 8, i(3) = 0, j(3) = 0; %e A323019 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. %e A323019 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. %e A323019 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. %e A323019 ... %e A323019 List of the multiplicative groups of Gaussian integers modulo members of this sequence: %e A323019 a(0) = 1: the trivial group; %e A323019 a(1) = 2: C_2; %e A323019 a(2) = 4: C_2 X C_4; %e A323019 a(3) = 8: C_2 X C_4 X C_4; %e A323019 a(4) = 20: C_2 X C_4 X C_4 X C_4; %e A323019 a(5) = 40: C_2 X C_4 X C_4 X C_4 X C_4; %e A323019 a(6) = 120: C_2 X C_4 X C_4 X C_4 X C_4 X C_8; %e A323019 a(7) = 520: C_2 X C_4 X C_4 X C_4 X C_4 X C_12 X C_12; %e A323019 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; %e A323019 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; %e A323019 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; %e A323019 ... %o A323019 (PARI) %o A323019 p(n) = my(i=0, k=0); while(i