A064279 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.

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

Offset

1,4

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

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

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

Mathematica

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 *)

Prog

(PARI)
A007434(n) = if(n<1, 0, sumdiv(n, d, d^2*moebius(n/d)));
a(n) = n^2 - A007434(n);  /* Joerg Arndt, Apr 14 2013 */

Crossrefs

Cf. A007434, A051953.

Sequence in context: A019304 A072869 A369905 * A369908 A078710 A175763

Adjacent sequences: A064276 A064277 A064278 * A064280 A064281 A064282

Keyword

nonn

Author

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

Extensions

More terms from Vladeta Jovovic, Sep 25 2001

Terms a(65..74) from Antti Karttunen, Jan 22 2025

Status

approved