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

1,1

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)

Links

Table of n, a(n) for n=1..30.

Wikipedia, Multiplicative group of integers modulo n

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

Cf. A057635, A289626.

Sequence in context: A040589 A040590 A040588 * A040591 A040592 A040593

Adjacent sequences: A303709 A303710 A303711 * A303713 A303714 A303715

Keyword

nonn,hard,more

Author

Jianing Song, Apr 29 2018

Extensions

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

Status

approved