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

8, 12, 63, 126, 513, 1026, 2107, 4214, 12625, 25250, 26533, 39609, 53066, 79218, 355023, 710046, 3190833, 4457713, 6381666, 8915426, 19854847, 38463283, 39709694, 76926566, 242138449, 370634743, 484276898, 516465451, 574336561, 701607583, 741269486, 1032930902, 1148673122, 1380336193, 1403215166, 2324581983, 2760672386, 4649163966, 4882890625, 6174434113, 9765781250

Offset: 1

Author

Jianing Song, Jun 19 2018

Keywords

• nonn

Comments

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.

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)).

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.

Numbers k such that A046072(k) = 2 and A316089(k) = 1. - Jianing Song, Sep 15 2018

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

Examples

			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.
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.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..287 (all terms below 10^16)
• Wikipedia, Multiplicative group of integers modulo n.

Crossrefs

Cf. A114874.

Odd terms are given by A307527.

Programs

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

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->(iJianing Song, Apr 13 2019

Formula

A302257(a(n)) = A258615(a(n))/2.

Extensions

Missing a(40) inserted by Jianing Song, Apr 20 2019