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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.
3
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
1,1
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
LINKS
Jianing Song, Table of n, a(n) for n = 1..287 (all terms below 10^16)
FORMULA
A302257(a(n)) = A258615(a(n))/2.
EXAMPLE
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.
PROG
(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->(i<nn), t)); t \\ Jianing Song, Apr 13 2019
CROSSREFS
Cf. A114874.
Odd terms are given by A307527.
Sequence in context: A212815 A298901 A368125 * A069186 A166625 A370651
KEYWORD
nonn
AUTHOR
Jianing Song, Jun 19 2018
EXTENSIONS
Missing a(40) inserted by Jianing Song, Apr 20 2019
STATUS
approved