cp's OEIS Frontend

A350638 Numbers of the form p^2*q, with odd primes p > q, such that q divides p-1.

%I A350638 #49 Jan 11 2022 22:13:28
%S A350638 147,507,605,1083,2883,4107,4805,5547,5819,5887,8405,11163,12943,
%T A350638 13467,15987,18605,18723,25205,28227,31827,35287,35643,36517,48387,
%U A350638 49379,50807,51005,57963,68403,73947,79707,81133,85805,87131,89383,98283,100949,111747,112903
%N A350638 Numbers of the form p^2*q, with odd primes p > q, such that q divides p-1.
%C A350638 For these terms m there are precisely (q+9)/2 groups of order m.
%C A350638 Only two of these groups are abelian: C_{p^2*q} and (C_p X C_p) X C_q. The (q+5)/2 groups that are nonabelian are C_{p^2} : C_q and the (q+3)/2 semidirect products of the form (C_p X C_p) : C_q that are not isomorphic, where C means cyclic groups of the stated order, the symbols X and : mean direct and semidirect products respectively.
%D A350638 Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.
%H A350638 N. S. Wedd, <a href="http://www.weddslist.com/groups/misc/147.html">Groups of orders 147</a>.
%e A350638 147 = 7^2 * 3, 3 and 7 are odd primes, 3 divides 7-1 = 6, hence 147 is a term.
%t A350638 q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; e == {1, 2} && Divisible[p[[2]] - 1, p[[1]]]]; Select[Range[1, 120000, 2], q] (* _Amiram Eldar_, Jan 11 2022 *)
%o A350638 (Python)
%o A350638 from sympy import integer_nthroot, primerange
%o A350638 def aupto(limit):
%o A350638     aset, maxp = set(), integer_nthroot(limit**2, 3)[0]
%o A350638     for p in primerange(5, maxp+1):
%o A350638         pp = p*p
%o A350638         for q in primerange(3, min(p, limit//pp+1)):
%o A350638             if (p-1)%q == 0:
%o A350638                 aset.add(pp*q)
%o A350638     return sorted(aset)
%o A350638 print(aupto(113000)) # _Michael S. Branicky_, Jan 10 2022
%o A350638 (PARI) isok(m) = if (m%2, my(f=factor(m)); if (f[, 2] == [1, 2]~, my(p=f[1, 1], q=f[2, 1]); ((q-1) % p) == 0)); \\ _Michel Marcus_, Jan 11 2022
%Y A350638 Other subsequences of A054753 linked with order of groups: A079704, A143928, A349495, A350115, A350245, A350332, A350421, A350422.
%K A350638 nonn
%O A350638 1,1
%A A350638 _Bernard Schott_, Jan 10 2022
%E A350638 More terms from _Michael S. Branicky_, Jan 10 2022