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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 (list; graph; refs; listen; history; text; internal format)
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)

Wikipedia, Multiplicative group of integers modulo n.

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: A286360 A212815 A298901 * A069186 A166625 A038290

Adjacent sequences:  A305233 A305234 A305235 * A305237 A305238 A305239

KEYWORD

nonn

AUTHOR

Jianing Song, Jun 19 2018

EXTENSIONS

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

STATUS

approved

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Last modified October 15 04:33 EDT 2019. Contains 328026 sequences. (Running on oeis4.)