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A350322
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Abelian orders m for which there exist exactly 2 groups of order m.
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7
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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Jianing Song, Table of n, a(n) for n = 1..10000
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EXAMPLE
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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}.
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PROG
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(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)[1], 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)
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Jianing Song, Dec 25 2021
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STATUS
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approved
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