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
Jianing Song, Table of n, a(n) for n = 1..160
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Jianing Song, List of k such that U(k) = U(a(n)) for n = 1..160.
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) a(n) = my(t=1); while(#select(x->znstar(x)[2]==znstar(t)[2], invphi(eulerphi(t)))!=n, t++); t \\ Jianing Song, Jan 11 2026 using Max Alekseyev's invphi.gp
CROSSREFS
KEYWORD
nonn,hard
AUTHOR
Jianing Song, Apr 29 2018
EXTENSIONS
a(21)-a(24) from Jianing Song, Oct 04 2018
STATUS
approved
