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A305236
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Numbers n such that the multiplicative group of integers modulo n is isomorphic to C_m X C_m, m > 1.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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
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LINKS
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FORMULA
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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