A064279 - OEIS

%I #22 Jan 22 2025 14:29:38

%S 0,1,1,4,1,12,1,16,9,28,1,48,1,52,33,64,1,108,1,112,57,124,1,192,25,

%T 172,81,208,1,324,1,256,129,292,73,432,1,364,177,448,1,612,1,496,297,

%U 532,1,768,49,700,297,688,1,972,145,832,369,844,1,1296,1,964,513,1024,193,1476,1,1168,537,1444,1,1728,1,1372

%N Number of ordered pairs a,b of elements in the cyclic group C_n such that the subgroup generated by the pair a,b is a proper subgroup of C_n.

%C For a prime p: a(p) = 1.

%H Antti Karttunen, <a href="/A064279/b064279.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = n^2 - A007434(n) = n^2 - J_2(n).

%F G.f.: -Sum_{k>=2} mu(k) * x^k * (1 + x^k) / (1 - x^k)^3. - _Ilya Gutkovskiy_, Sep 14 2021

%t Table[a=GroupElements[CyclicGroup[n]];n^2-Count[Flatten[Table[Table[GroupOrder[PermutationGroup[{a[[i]],a[[j]]}]],{i,1,n}],{j,1,n}]],n],{n,1,30}]  (* _Geoffrey Critzer_, Apr 14 2013 *)

%o (PARI)

%o A007434(n) = if(n<1, 0, sumdiv(n, d, d^2*moebius(n/d)));

%o a(n) = n^2 - A007434(n);  /* _Joerg Arndt_, Apr 14 2013 */

%Y Cf. A007434, A051953.

%K nonn

%O 1,4

%A Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Sep 24 2001

%E More terms from _Vladeta Jovovic_, Sep 25 2001

%E Terms a(65..74) from _Antti Karttunen_, Jan 22 2025