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A305236 Numbers n such that the multiplicative group of integers modulo n is isomorphic to C_m X C_m, m > 1.

%I A305236 #32 Apr 21 2019 04:46:21
%S A305236 8,12,63,126,513,1026,2107,4214,12625,25250,26533,39609,53066,79218,
%T A305236 355023,710046,3190833,4457713,6381666,8915426,19854847,38463283,
%U A305236 39709694,76926566,242138449,370634743,484276898,516465451,574336561,701607583,741269486,1032930902,1148673122,1380336193,1403215166,2324581983,2760672386,4649163966,4882890625,6174434113,9765781250
%N A305236 Numbers n such that the multiplicative group of integers modulo n is isomorphic to C_m X C_m, m > 1.
%C A305236 Note that 24 is only number k such that the multiplicative group of integers modulo k is isomorphic to C_m X C_m X C_m, m > 1.
%C A305236 The number of elements in the multiplicative group of integers modulo a(n) of order d is A007434(d), whenever d is divisible by A002322(a(n)).
%C A305236 The corresponding m (=A002322(a(n))) are 2, 2, 6, 6, 18, 18, 42, 42, 100, 100, 156, 162, 156, 162, 486, 486, 1458, 2028, 1458, 2028, ... Each term in A114874, except for those of the form 2^k, k >= 2, occurs exactly twice in this list.
%C A305236 Numbers k such that A046072(k) = 2 and A316089(k) = 1. - _Jianing Song_, Sep 15 2018
%C A305236 Except for 8 and 12, these are numbers of the form p^e*((p-1)*p^(e-1) + 1) or 2*p^e*((p-1)*p^(e-1) + 1) where p is an odd prime and (p-1)*p^(e-1) + 1 is prime. - _Jianing Song_, Apr 13 2019
%H A305236 Jianing Song, <a href="/A305236/b305236.txt">Table of n, a(n) for n = 1..287</a> (all terms below 10^16)
%H A305236 Wikipedia, <a href="http://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n">Multiplicative group of integers modulo n</a>.
%F A305236 A302257(a(n)) = A258615(a(n))/2.
%e A305236 The multiplicative group of integers modulo 63 is isomorphic to C_6 X C_6. There are A007434(1) = 1 element of order 1, A007434(2) = 3 elements of order 2, A007434(3) = 8 elements of order 3, A007434(6) = 24 elements of order 6 modulo 63.
%e A305236 The multiplicative group of integers modulo 513 is isomorphic to C_18 X C_18. There are A007434(1) = 1 element of order 1, A007434(2) = 3 elements of order 2, A007434(3) = 8 elements of order 3, A007434(6) = 24 elements of order 6, A007434(9) = 72 elements of order 9, A007434(18) = 216 elements of order 18 modulo 513.
%o A305236 (PARI) for(n=1,10^7,if(#znstar(n)[2]==2 && znstar(n)[2][1]==znstar(n)[2][2], print1(n, ", "))) \\ _Jianing Song_, Sep 15 2018
%o A305236 (PARI) the_first_entries(nn) = my(u=[]); for(n=2, sqrt(nn), my(v=factor(n), d=#v[, 1], p=v[d, 1], e=v[d, 2]); if(isprime(n+1) && p!=2 && n==(p-1)*p^e, u=concat(u, [(n+1)*p^(e+1)]))); t=concat([8, 12], concat(u, 2*u)); t=vecsort(select(i->(i<nn), t)); t \\ _Jianing Song_, Apr 13 2019
%Y A305236 Cf. A114874.
%Y A305236 Odd terms are given by A307527.
%K A305236 nonn
%O A305236 1,1
%A A305236 _Jianing Song_, Jun 19 2018
%E A305236 Missing a(40) inserted by _Jianing Song_, Apr 20 2019