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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.

%I A303712 #54 May 12 2024 18:11:29
%S A303712 24,1,3,7,55,129,35,104,407,707,143,371,899,665,1144,1771,385,3003,
%T A303712 3451,5005,7049,8041,7579,12243,4081,5291,3857,9361,2717,2233
%N 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.
%C A303712 Conjecture: a(n) > 0 for all n.
%C A303712 Other known terms: a(35) = 8855, a(39) = 6149. [Corrected by _Jianing Song_, Oct 04 2018]
%C A303712 From _Jianing Song_, Oct 04 2018: (Start)
%C A303712 a(32) = 9269, a(33) = 7315, a(37) = 15953, a(52) = 16555, a(59) = 17081.
%C A303712 a(31), a(34), a(36), a(38) etc. > 2*10^4 (if not equal to 0). (End)
%H A303712 Wikipedia, <a href="http://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n">Multiplicative group of integers modulo n</a>
%e A303712 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.
%e A303712 U(1) and U(2) are both isomorphic to the trivial group.
%e A303712 U(3), U(4) and U(6) are isomorphic to C_2.
%e A303712 U(7), U(9), U(14) and U(18) are isomorphic to C_6.
%e A303712 U(55), U(75), U(100), U(110) and U(150) are isomorphic to C_2 x C_20.
%e A303712 U(129), U(147), U(172), U(196), U(258) and U(294) are isomorphic to C_2 x C_42.
%e A303712 U(35), U(39), U(45), U(52), U(70), U(78) and U(90) are isomorphic to C_2 x C_12.
%e A303712 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.
%o A303712 (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
%o A303712 a(n) = if(n==2, 1, my(t=3); while(b(t)!=n, t++); t) \\ _Jianing Song_, Oct 04 2018
%Y A303712 Cf. A057635, A289626.
%K A303712 nonn,hard,more
%O A303712 1,1
%A A303712 _Jianing Song_, Apr 29 2018
%E A303712 a(21)-a(24) from _Jianing Song_, Oct 04 2018