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A350343 Square numbers k that are abelian orders.

%I A350343 #17 Jan 05 2022 07:19:17
%S A350343 1,4,9,25,49,121,169,289,361,529,841,961,1225,1369,1681,1849,2209,
%T A350343 2809,3481,3721,4225,4489,5041,5329,5929,6241,6889,7225,7921,9409,
%U A350343 10201,10609,11449,11881,12769,13225,14161,16129,17161,17689,18769,19321,20449,22201,22801
%N A350343 Square numbers k that are abelian orders.
%C A350343 k must be the square of a squarefree number. Actually, k must be the square of a cyclic number (A003277).
%C A350343 Number of the form (p_1*p_2*...*p_r)^2 where the p_i are distinct primes and no (p_j)^2-1 is divisible by any p_i.
%C A350343 The smallest term with exactly n distinct prime factors is given by A350341.
%C A350343 From the term 25 on, no term can be divisible by 2 or 3.
%H A350343 Jianing Song, <a href="/A350343/b350343.txt">Table of n, a(n) for n = 1..10000</a>
%F A350343 a(n) = A350342(n)^2.
%e A350343 For primes p, p^2 is a term since every group of order p^2 is abelian. Such group is isomorphic to either C_{p^2} or C_p X C_p.
%e A350343 For primes p, q, if p^2 !== 1 (mod q), q^2 !== 1 (mod p), then p^2*q^2 is a term since every group of that order is abelian. Such group is isomorphic to C_{p^2*q^2}, C_p X C_{p*q^2}, C_q X C_{p^2*q} or C_{p*q} X C_{p*q}.
%o A350343 (PARI) isA051532(n) = my(f=factor(n), v=vector(#f[, 1])); for(i=1, #v, if(f[i, 2]>2, return(0), v[i]=f[i, 1]^f[i, 2])); for(i=1, #v, for(j=i+1, #v, if(v[i]%f[j, 1]==1 || v[j]%f[i, 1]==1, return(0)))); 1 \\ _Charles R Greathouse IV_'s program for A051532
%o A350343 isA350343(n) = issquare(n) && isA051532(n)
%Y A350343 Cf. A051532 (abelian orders), A003277 (cyclic numbers), A350342, A350341.
%Y A350343 A350152 = A350322 U A350323 is a subsequence. A350345 is the subsequence of squares of composite numbers.
%K A350343 nonn
%O A350343 1,2
%A A350343 _Jianing Song_, Dec 25 2021