A051532 -id:A051532 - cp's OEIS frontend

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

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

Jianing Song, Dec 25 2021

nonn

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.

			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}.

Jianing Song, Table of n, a(n) for n = 1..10000

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.

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)

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)

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

A059806 Minimal size of the center of G where G is a finite group of order n.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 7, 2, 9, 1, 11, 1, 13, 1, 15, 2, 17, 1, 19, 1, 1, 1, 23, 1, 25, 1, 3, 2, 29, 1, 31, 2, 33, 1, 35, 1, 37, 1, 1, 2, 41, 1, 43, 2, 45, 1, 47, 1, 49, 1, 51, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 3, 2, 65, 1, 67, 1, 69, 1, 71, 1, 73, 1, 1, 2, 77, 1

Offset: 1

Noam Katz (noamkj(AT)hotmail.com), Feb 24 2001

nonn

a(n) = n if and only if n belongs to sequence A051532 - Avi Peretz (njk(AT)netvision.net.il), Feb 27 2001

a(n) = 1 if and only if n occurs in A060702. - Eric M. Schmidt, Aug 27 2012

			a(6) = 1 because the symmetric group S_3 has trivial center.

Eric M. Schmidt, Table of n, a(n) for n = 1..2000
MathOverflow, Center of p-groups

Cf. A059807, A051532.

A059806 := function(n) local min, fact, i; if (n mod 6 = 0) then return 1; fi; if (IsPrimePowerInt(n)) then fact := Factors(n); if (Length(fact) <> 2) then return fact[1]; fi; fi; min := n; for i in [1..NumberSmallGroups(n)] do min := Minimum(min, Size(Center(SmallGroup(n, i)))); if (min = 1) then break; fi; od; return min; end; # Eric M. Schmidt, Aug 27 2012

For prime p and m >= 3, a(p^m) = p. - Eric M. Schmidt, Aug 27 2012

More terms from Eric M. Schmidt, Aug 27 2012

A350332 Numbers p^2*q, p < q odd primes such that p does not divide q-1.

Original entry on oeis.org

45, 99, 153, 175, 207, 261, 325, 369, 423, 425, 475, 477, 531, 539, 575, 637, 639, 725, 747, 801, 833, 909, 925, 931, 963, 1017, 1075, 1127, 1175, 1179, 1233, 1325, 1341, 1475, 1503, 1519, 1557, 1573, 1611, 1675, 1719, 1773, 1813, 1825, 1975, 2009, 2043, 2057

Offset: 1

Bernard Schott, Dec 25 2021

nonn

For these terms m, there are precisely 2 groups of order m, so this is a subsequence of A054395.

The 2 groups are abelian; they are C_{p^2*q} and (C_p X C_p) X C_q, where C means cyclic groups of the stated order and the symbol X means direct product.

			99 = 3^2 * 11, 3 and 11 are odd and 3 does not divide 11-1 = 10, hence 99 is a term.
175 = 5^2 * 7, 5 and 7 are odd and 5 does not divide 7-1 = 6, hence 115 is another term.

Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.

Subsequence of A051532, A054395, A054753 and of A060687.

Other subsequences of A054753 linked with order of groups: A079704, A143928, A349495, A350115, A350245.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; e == {2, 1} && ! Divisible[p[[2]] - 1, p[[1]]]]; Select[Range[2000], q] (* Amiram Eldar, Dec 25 2021 *)

isok(m) = my(f=factor(m)); if (f[, 2] == [2, 1]~, my(p=f[1, 1], q=f[2, 1]); ((q-1) % p)); \\ Michel Marcus, Dec 25 2021

from sympy import integer_nthroot, primerange
def aupto(limit):
    aset, maxp = set(), integer_nthroot(limit, 3)[0]
    for p in primerange(3, maxp+1):
        pp = p*p
        for q in primerange(p+1, limit//pp+1):
            if (q-1)%p != 0:
                aset.add(pp*q)
    return sorted(aset)
print(aupto(2060)) # Michael S. Branicky, Dec 25 2021

More terms from Michael S. Branicky, Dec 25 2021

A350323 Abelian orders m for which there exist at least 4 groups of order m.

Original entry on oeis.org

1225, 4225, 5929, 7225, 13225, 14161, 15925, 17689, 20449, 20825, 23275, 25921, 28175, 34225, 34969, 43681, 45325, 46225, 47089, 48841, 50575, 55225, 57575, 61009, 64925, 67081, 70225, 70805, 71825, 72275, 77077, 80275, 82075, 89401, 89425, 92575, 93925, 96775, 97175

Offset: 1

Jianing Song, Dec 25 2021

nonn

Abelian orders of the form (p_1)^2 * (p_2)^2 * ... * (p_r)^2 * q_1 * q_2 * ... * q_s, r >= 2, 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. Note that there are 2^r groups of such order.

No term can be divisible by 2 or 3.

			For primes p, q, if p^2 !== 1 (mod q) and q^2 !== 1 (mod p), then p^2*q^2 is a term since the 4 groups of that order are C_{p^2*q^2}, C_p X C_{p*q^2}, C_q X C_{p^2*q}, C_{p*q} X C_{p*q}.

Jianing Song, Table of n, a(n) for n = 1..10000

Equals A350152 \ A350322 = A051532 \ (A005117 U A060687).

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
isA350323(n) = isA051532(n) && (bigomega(n)-omega(n)>1)

A350340 a(n) is the smallest k such that k^2 is an abelian order with precisely 2^n groups.

Original entry on oeis.org

1, 2, 35, 595, 13685, 506345, 26836285, 1702480351, 80016576497, 5681176931287, 414725915983951, 40228413850443247, 4304440281997427429, 546663915813673283483, 75986284298100586404137, 10144780646398552482233711, 1511572316313384319852822939, 252432576824335181415421430813, 49729217634394030738838021870161

Offset: 0

Jianing Song, Dec 25 2021

If m is an abelian order, then m = (p_1)^2 * (p_2)^2 * ... * (p_r)^2 * q_1 * q_2 * ... * q_s, where p_1, p_2, ... p_r, q_1, q_2, ..., q_s are distinct primes such that (p_i)^2 !== 1 (mod p_j) for i != j, (p_i)^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.
a(n) is the smallest k with n distinct prime factors such that k^2 is an abelian order.
a(n) is the smallest number of the form p_1*p_2*...*p_n where the p_i are distinct primes and no (p_j)^2-1 is divisible by any p_i.
a(n) exists for all n.
Except for a(1) = 2, no term can be divisible by 2 or 3. Conjecture: lpf(a(n+1)) >= lpf(a(n)) for all n, where lpf = least prime factor. - David A. Corneth and Jianing Song, Jan 03 2022

			a(2) = 35 = 5*7 since the smallest k with 2 distinct prime factors such that k^2 is an abelian order is 35.
a(3) = 595 = 5*7*17 since the smallest k with 3 distinct prime factors such that k^2 is an abelian order is 595.

David A. Corneth, Table of n, a(n) for n = 0..30
David A. Corneth, PARI program

Cf. A051532 (abelian orders), A264907, A350341.

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
a(n) = for(k=1, oo, if(issquarefree(k) && omega(k)==n && isA051532(k^2), return(k)))

A350341(n) = a(n)^2.

a(7)-a(18) from David A. Corneth, Jan 02 2022

A350341 a(n) is the smallest abelian order with precisely 2^n groups.

Original entry on oeis.org

1, 4, 1225, 354025, 187279225, 256385259025, 720186192601225, 2898439345541083201, 6402652514300252791009, 32275771324587574319476369, 171997585388727183548489570401, 1618325280922534070007738367903009, 18528206141282092567518596574121550041, 298841436852738871021507444144006480611289

Offset: 0

Jianing Song, Dec 25 2021

If m is an abelian order, then m = (p_1)^2 * (p_2)^2 * ... * (p_r)^2 * q_1 * q_2 * ... * q_s, where p_1, p_2, ... p_r, q_1, q_2, ..., q_s are distinct primes such that (p_i)^2 !== 1 (mod p_j) for i != j, (p_i)^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.
a(n) is the smallest square number with n distinct prime factors that is an abelian order.
a(n) is the smallest number of the form (p_1*p_2*...*p_n)^2 where the p_i are distinct primes and no (p_j)^2-1 is divisible by any p_i.
a(n) exists for all n.
Except for a(1) = 4, no term can be divisible by 2 or 3. Conjecture: lpf(a(n+1)) >= lpf(a(n)) for all n, where lpf = least prime factor.

			a(2) = 1225 = 35^2 since the smallest square number with 2 distinct prime factors that is an abelian order is 1225.
a(3) = 354025 = 595^2 since the smallest square number with 3 distinct prime factors that is an abelian order is 354025.

David A. Corneth, Table of n, a(n) for n = 0..18

Cf. A051532 (abelian orders), A264907, A350340.

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
a(n) = for(k=1, oo, if(issquarefree(k) && omega(k)==n && isA051532(k^2), return(k^2)))

a(n) = A350340(n)^2.

a(7)-a(11) from David A. Corneth, Jan 02 2022

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

Jianing Song, Dec 25 2021

nonn

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.

			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}.

Jianing Song, Table of n, a(n) for n = 1..10000

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.

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)

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

A350422 Numbers of the form m = p^2*q for which there exist exactly 2 groups of order m.

Original entry on oeis.org

45, 99, 153, 175, 207, 245, 261, 325, 369, 423, 425, 475, 477, 531, 539, 575, 637, 639, 725, 747, 801, 833, 845, 847, 909, 925, 931, 963, 1017, 1075, 1127, 1175, 1179, 1233, 1325, 1341, 1445, 1475, 1503, 1519, 1557, 1573, 1611, 1675, 1719, 1773, 1813, 1825, 1859, 1975, 2009

Offset: 1

Bernard Schott, Jan 03 2022

nonn

Terms come from the union of terms of the form p^2*q with p < q in A350332 and terms of the same form with p > q in A350421, with p, q odd primes.
All terms are odd.
These 2 groups are abelian; they are C_{p^2*q} and (C_p X C_p) X C_q, where C means cyclic groups of the stated order and the symbol X means direct product.

			With p < q: 175 = 5^2 * 7, 5 and 7 are odd primes and 5 does not divide 7-1 = 6, hence 175 is a term (see A350332).
With p > q: 245 = 7^2 * 5, 5 and 7 are odd primes, 5 does not divide 7-1 = 6 and does not divide 7+1 = 8, hence 245 is a term (see A350421).

Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.

Disjoint union of A350332 (pA350421 (p>q).
Intersection of A054395 and A054753.
Subsequence of A051532, A060687 and A350322.
Other subsequences of A054753 linked with order of groups: A079704, A143928, A349495, A350115, A350245, A350638.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; (e == {1, 2} && ! Or @@ Divisible[p[[2]] + {-1, 1}, p[[1]]]) || (e == {2, 1} && ! Divisible[p[[2]] - 1, p[[1]]])]; Select[Range[1, 2000, 2], q] (* Amiram Eldar, Jan 03 2022 *)

isoka(f) = if (f[, 2] == [2, 1]~, my(p=f[1, 1], q=f[2, 1]); ((q-1) % p)); \\ A350332
isokb(f) = if (f[, 2] == [1, 2]~, my(p=f[2, 1], q=f[1, 1]); ((p-1) % q) && ((p+1) % q)); \\ A350421
isok(m) = my(f=factor(m)); isoka(f) || isokb(f); \\ Michel Marcus, Jan 09 2022

A059807 Maximal size of the commutator subgroup of G where G is a finite group of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 2, 1, 5, 1, 4, 1, 7, 1, 4, 1, 9, 1, 5, 7, 11, 1, 12, 1, 13, 3, 7, 1, 15, 1, 8, 1, 17, 1, 9, 1, 19, 13, 10, 1, 21, 1, 11, 1, 23, 1, 24, 1, 25, 1, 13, 1, 27, 11, 14, 19, 29, 1, 60, 1, 31, 7, 16, 1, 33, 1, 17, 1, 35, 1, 36, 1, 37, 25, 19, 1

Offset: 1

Noam Katz (noamkj(AT)hotmail.com), Feb 24 2001

nonn

a(n) = 1 iff n belongs to sequence A051532. - Avi Peretz (njk(AT)netvision.net.il), Feb 27 2001

			a(6) = 3 because the commutator subgroup of the symmetric group S_3 is the group Z_3.

Eric M. Schmidt, Table of n, a(n) for n = 1..2000
MathOverflow, Center of p-groups

Cf. A059806, A051532.

A059807 := function(n) local max, fact, i; if (IsPrimePowerInt(n)) then fact := Factors(n); if (Length(fact) >= 2) then return n/fact[1]^2; fi; fi; max := 1; for i in [1..NumberSmallGroups(n)] do max := Maximum(max, Size(DerivedSubgroup(SmallGroup(n, i)))); od; return max; end; # Eric M. Schmidt, Sep 20 2012

For prime p and m >= 2, a(p^m) = p^(m - 2). - Eric M. Schmidt, Sep 20 2012

More terms from Eric M. Schmidt, Sep 20 2012

A083573 Maximal number of subgroups in a non-Abelian group with n elements, or zero if there are no non-Abelian groups of order n.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 0, 10, 0, 8, 0, 16, 0, 10, 0, 35, 0, 28, 0, 22, 10, 14, 0, 54, 0, 16, 19, 28, 0, 28, 0, 158, 0, 20, 0, 78, 0, 22, 16, 76, 0, 36, 0, 40, 0, 26, 0, 236, 0, 64, 0, 46, 0, 212, 14, 98, 22, 32, 0, 80, 0, 34, 36, 937, 0, 52, 0, 58, 0, 52, 0, 272

Offset: 1

Victoria A. Sapko (vsapko(AT)canes.gsw.edu), Jun 13 2003

nonn

A group G is non-Abelian iff there are two elements x,y such that xy != yx. Then and are nontrivial subgroups whose order divides the order of G which therefore cannot be prime (neither the square of a prime: there are only two nonisomorphic groups of that order which are both abelian; see A051532 for more). This also implies that a(n) >= 2+2+2 = 6 for all nonzero elements of this sequence and for even n=2m>4 there is the non-Abelian dihedral group D_m with A007503(m)=sigma(m)+tau(m)=A000005(m)+A000203(m), providing a lower bound. - M. F. Hasler, Dec 03 2007

			a(6)=6 because the only non-Abelian group with 6 elements is S_3 with 6 subgroups.

Eric M. Schmidt, Table of n, a(n) for n = 1..511

Cf. A018216, A061034.

Cf. A051532, A060689, A007503.

A083573 := function(n) local max, grp, i; max := 0; for i in [1..NumberSmallGroups(n)] do grp := SmallGroup(n, i); if (not IsAbelian(grp)) then max := Maximum(max, Sum(ConjugacyClassesSubgroups(grp), Size)); fi; od; return max; end; # Eric M. Schmidt, Sep 07 2012

a(n) = 0 <=> A060689(n)=0 <=> n is in A051532 ; otherwise a(n) >= 6 and a(2n) >= A007503(n). - M. F. Hasler, Dec 03 2007

More terms from Eric M. Schmidt, Sep 07 2012

cp's OEIS Frontend

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

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A059806 Minimal size of the center of G where G is a finite group of order n.

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A350332 Numbers p^2*q, p < q odd primes such that p does not divide q-1.

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A350323 Abelian orders m for which there exist at least 4 groups of order m.

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A350340 a(n) is the smallest k such that k^2 is an abelian order with precisely 2^n groups.

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A350341 a(n) is the smallest abelian order with precisely 2^n groups.

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A350343 Square numbers k that are abelian orders.

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