The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A327925 Irregular table read by rows: T(m,n) is the number of non-isomorphic groups G such that G is the semidirect product of C_m and C_n, where C_m is a normal subgroup of G and C_n is a subgroup of G, 1 <= n <= A002322(m). 2
 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 2, 2, 1, 4, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 2, 2, 2, 1, 2, 1, 4, 1, 4, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 1, 6, 1, 2, 2, 2, 1, 4, 1, 4, 1, 6, 1, 4, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 3, 2, 1, 4, 1, 2, 2, 2, 1, 6 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS The semidirect product of C_m and C_n has group representation G = , where r is any number such that r^n == 1 (mod m). Two groups G = and G' = are isomorphic if and only if there exists some k, gcd(k,n) = 1 such that r^k == s (mod m), in which case f(x^i*y^j) = x^i*y^(k*j) is an isomorphic mapping from G to G'. Given m, T(m,n) only depends on the value of gcd(n,psi(m)), psi = A002322 (Carmichael lambda). So each row of A327924 is periodic with period psi(m), so we have this for an alternative version. Every number k occurs in the table. By Dirichlet's theorem on arithmetic progressions, there exists a prime p such that p == 1 (mod 2^(k-1)), then T(p,2^(k-1)) = d(gcd(2^(k-1),p-1)) = k (see the formula below). For example, T(5,4) = 3, T(17,8) = 4, T(17,16) = 5, T(97,32) = 6, T(193,64) = 7, ... Row m and Row m' are the same if and only if (Z/mZ)* = (Z/m'Z)*, where (Z/mZ)* is the multiplicative group of integers modulo m. The if part is clear; for the only if part, note that the two sequences {(number of x in (Z/mZ)* such that x^n = 1)}_{n>=1} and {T(m,n)}_{n>=1} determine each other, and the structure of a finite abelian group G is uniquely determined by the sequence {(number of x in G such that x^n = 1)}_{n>=1}. - Jianing Song, May 16 2022 LINKS Jianing Song, Table of n, a(n) for n = 1..8346 (the first 200 rows) Math Overflow, When are two semidirect products of two cyclic groups isomorphic FORMULA T(m,n) = Sum_{d|n} (number of elements x such that ord(x,m) = d)/phi(d), where ord(x,m) is the multiplicative order of x modulo m, phi = A000010. Equivalently, T(m,n) = Sum_{d|gcd(n,psi(m))} (number of elements x such that ord(x,m) = d)/phi(d). - Jianing Song, May 16 2022 For odd primes p, T(p^e,n) = d(gcd(n,(p-1)*p^(e-1))) = A051194((p-1)*p^(e-1),n), d = A000005; for e >= 3, T(2^e,n) = 2*(v2(n)+1) for even n and 1 for odd n, where v2 is the 2-adic valuation. EXAMPLE Table starts m = 1: 1; m = 2: 1; m = 3: 1, 2; m = 4: 1, 2; m = 5: 1, 2, 1, 3; m = 6: 1, 2; m = 7: 1, 2, 2, 2, 1, 4; m = 8: 1, 4; m = 9: 1, 2, 2, 2, 1, 4; m = 10: 1, 2, 1, 3; m = 11: 1, 2, 1, 2, 2, 2, 1, 2, 1, 4; m = 12: 1, 4; m = 13: 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 1, 6; m = 14: 1, 2, 2, 2, 1, 4; m = 15: 1, 4, 1, 6; m = 16: 1, 4, 1, 6; m = 17: 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5; m = 18: 1, 2, 2, 2, 1, 4; m = 19: 1, 2, 2, 2, 1, 4, 1, 2, 3, 2, 1, 4, 1, 2, 2, 2, 1, 6; m = 20: 1, 4, 1, 6; Example shows that T(21,6) = 6: The semidirect product of C_21 and C_6 has group representation G = , where r = 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20. Since 2^5 == 11 (mod 21), 4^5 == 16 (mod 21), 5^5 == 17 (mod 21), 10^5 == 19 (mod 21), there are actually four pairs of isomorphic groups, giving a total of 8 non-isomorphic groups. PROG (PARI) numord(n, q) = my(v=divisors(q), r=znstar(n)); sum(i=1, #v, prod(j=1, #r, gcd(v[i], r[j]))*moebius(q/v[i])) T(m, n) = my(u=divisors(n)); sum(i=1, #u, numord(m, u[i])/eulerphi(u[i])) Row(m) = my(l=if(m>2, znstar(m), 1), R=vector(l, n, T(m, n))); R CROSSREFS Cf. A327925, A002322, A000010, A000005. Cf. also A060594, A060839, A073103, A319099, A319100, A319101, A051194. Sequence in context: A060938 A087942 A359237 * A320012 A359508 A352825 Adjacent sequences: A327922 A327923 A327924 * A327926 A327927 A327928 KEYWORD nonn,tabf,changed AUTHOR Jianing Song, Sep 30 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 1 14:06 EDT 2023. Contains 365826 sequences. (Running on oeis4.)