cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A350322 Abelian orders m for which there exist exactly 2 groups of order m.

Original entry on oeis.org

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
Offset: 1

Views

Author

Jianing Song, Dec 25 2021

Keywords

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.

Examples

			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}.

Links

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.

Programs

  • 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)
  • Python
    def is_ok(m):
    f = factorint(m)
    return (
        sum(f.values()) == len(f) + 1 and
        all((q - 1) % p > 0 for p in f for q in f) and
        (m := next(p for p, e in f.items() if e == 2) ** 2 - 1) and
        all(m % q > 0 for q in f)) # David Radcliffe, Jul 30 2025

A350343 Square numbers k that are abelian orders.

Original entry on oeis.org

1, 4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1225, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4225, 4489, 5041, 5329, 5929, 6241, 6889, 7225, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 13225, 14161, 16129, 17161, 17689, 18769, 19321, 20449, 22201, 22801
Offset: 1

Views

Author

Jianing Song, Dec 25 2021

Keywords

Comments

k must be the square of a squarefree number. Actually, k must be the square of a cyclic number (A003277).
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.
The smallest term with exactly n distinct prime factors is given by A350341.
From the term 25 on, no term can be divisible by 2 or 3.

Examples

			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.
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}.

Links

Crossrefs

Cf. A051532 (abelian orders), A003277 (cyclic numbers), A350342, A350341.
A350152 = A350322 U A350323 is a subsequence. A350345 is the subsequence of squares of composite numbers.

Programs

  • 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
isA350343(n) = issquare(n) && isA051532(n)

Formula

a(n) = A350342(n)^2.

A350345 Squares of composite numbers k that are abelian orders.

Original entry on oeis.org

1225, 4225, 5929, 7225, 13225, 14161, 17689, 20449, 25921, 34225, 34969, 43681, 46225, 47089, 48841, 55225, 61009, 67081, 70225, 89401, 101761, 104329, 108241, 112225, 116281, 133225, 137641, 142129, 152881, 162409, 165649, 170569, 172225, 182329, 190969
Offset: 1

Views

Author

Jianing Song, Dec 25 2021

Keywords

Comments

Square numbers k that are abelian orders with at least 4 groups.
Number of the form (p_1*p_2*...*p_r)^2 where r > 1, the p_i are distinct primes and no (p_j)^2-1 is divisible by any p_i.
The smallest square number k that is an abelian order with at least 8 groups is A350341(3) = 354025.
No term can be divisible by 2 or 3.

Examples

			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}.

Links

Crossrefs

Cf. A051532 (abelian orders), A050384, A350341.
Equals A350343 \ ({1} U A001248).
A350323 is a subsequence.

Programs

  • 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
isA350345(n) = issquare(n) && (n>1) && !isprime(sqrtint(n)) && isA051532(n^2)

Formula

a(n) = A350344(n)^2.
Showing 1-3 of 3 results.

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