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A303712
a(n) is the smallest number such that there are exactly n numbers k (including a(n) itself) such that U(k) is isomorphic to U(a(n)) (or 0 if no such number exists). Here U(k) is the multiplicative group of integers modulo k.
1
24, 1, 3, 7, 55, 129, 35, 104, 407, 707, 143, 371, 899, 665, 1144, 1771, 385, 3003, 3451, 5005, 7049, 8041, 7579, 12243, 4081, 5291, 3857, 9361, 2717, 2233
OFFSET
1,1
COMMENTS
Conjecture: a(n) > 0 for all n.
Other known terms: a(35) = 8855, a(39) = 6149. [Corrected by Jianing Song, Oct 04 2018]
From Jianing Song, Oct 04 2018: (Start)
a(32) = 9269, a(33) = 7315, a(37) = 15953, a(52) = 16555, a(59) = 17081.
a(31), a(34), a(36), a(38) etc. > 2*10^4 (if not equal to 0). (End)
EXAMPLE
U(24) is isomorphic to C_2 x C_2 x C_2 and there is no other number k such that U(k) is isomorphic to U(24), so a(1) = 24.
U(1) and U(2) are both isomorphic to the trivial group.
U(3), U(4) and U(6) are isomorphic to C_2.
U(7), U(9), U(14) and U(18) are isomorphic to C_6.
U(55), U(75), U(100), U(110) and U(150) are isomorphic to C_2 x C_20.
U(129), U(147), U(172), U(196), U(258) and U(294) are isomorphic to C_2 x C_42.
U(35), U(39), U(45), U(52), U(70), U(78) and U(90) are isomorphic to C_2 x C_12.
U(104), U(105), U(112), U(140), U(144), U(156), U(180) and U(210) are isomorphic to C_2 x C_2 x C_12.
PROG
(PARI) b(n) = my(i=0, search_max = A057635(eulerphi(n))); for(j=eulerphi(n)+1, search_max, if(znstar(j)[2]==znstar(n)[2], i++)); i \\ search_max is the largest k such that phi(k) = phi(n). See A057635 for its program
a(n) = if(n==2, 1, my(t=3); while(b(t)!=n, t++); t) \\ Jianing Song, Oct 04 2018
CROSSREFS
Sequence in context: A040589 A040590 A040588 * A040591 A040592 A040593
KEYWORD
nonn,hard,more
AUTHOR
Jianing Song, Apr 29 2018
EXTENSIONS
a(21)-a(24) from Jianing Song, Oct 04 2018
STATUS
approved