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 A350322 Abelian orders m for which there exist exactly 2 groups of order m. 7
 4, 9, 25, 45, 49, 99, 121, 153, 169, 175, 207, 245, 261, 289, 325, 361, 369, 423, 425, 475, 477, 529, 531, 539, 575, 637, 639, 725, 747, 765, 801, 833, 841, 845, 847, 909, 925, 931, 961, 963, 1017, 1035, 1075, 1127, 1175, 1179, 1233, 1305, 1325, 1341, 1369, 1445, 1475 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Abelian orders of the form p^2 * q_1 * q_2 * ... * q_s, where p, q_1, q_2, ..., q_s are distinct primes such that p^2 !== 1 (mod q_j), q_i !== 1 (mod p_j), q_i !== 1 (mod q_j) for i != j. In this case there are 2^r groups of order m. Note that the smallest abelian order with precisely 2^n groups must be the square of a squarefree number. Except for a(1) = 4, all terms are odd. The terms that are divisible by 3 are of the form 9 * q_1 * q_2 * ... * q_s, where q_i are distinct primes congruent to 5 modulo 6, q_i !== 1 (mod q_j) for i != j. LINKS Jianing Song, Table of n, a(n) for n = 1..10000 EXAMPLE For primes p, p^2 is a term since the 2 groups of that order are C_{p^2} and C_p X C_p. For primes p, q, if p^2 !== 1 (mod q) and q !== 1 (mod p), then p^2*q is a term since the 2 groups of that order are C_{p^2*q} and C_p X C_{p*q}. PROG (PARI) isA054395(n) = { my(p=gcd(n, eulerphi(n)), f); if (!isprime(p), return(0)); if (n%p^2 == 0, return(1 == gcd(p+1, n))); f = factor(n); 1 == sum(k=1, matsize(f), f[k, 1]%p==1); } \\ Gheorghe Coserea's program for A054395 isA350322(n) = isA054395(n) && (bigomega(n)-omega(n)==1) (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 isA350322(n) = isA051532(n) && (bigomega(n)-omega(n)==1) CROSSREFS Equals A060687 INTERSECT A051532 = A054395 INTERSECT A051532 = A054395 INTERSECT A060687 = A054395 INTERSECT A013929. Equals A350152 \ A350323. Equals A054395 \ A350586. Subsequence of A350152. A001248 and A350332 are subsequences. Sequence in context: A046451 A341790 A350152 * A158142 A158143 A082200 Adjacent sequences: A350319 A350320 A350321 * A350323 A350324 A350325 KEYWORD nonn AUTHOR Jianing Song, Dec 25 2021 STATUS approved

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Last modified March 24 04:28 EDT 2023. Contains 361454 sequences. (Running on oeis4.)