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.

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

Author

Jianing Song, Apr 29 2018

Keywords

• nonn
• hard
• more

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)

Examples

			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.
		

Links

• Wikipedia, Multiplicative group of integers modulo n

Crossrefs

Cf. A057635, A289626.

Programs

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

Extensions

a(21)-a(24) from Jianing Song, Oct 04 2018