A051532 -id:A051532 - cp's OEIS frontend

A350152 Abelian orders m for which there exist at least 2 groups with 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, 1225, 1233, 1305, 1325, 1341, 1369, 1445, 1475

Offset: 1

Bernard Schott, Dec 18 2021

nonn

This sequence lists the abelian orders when there is an abelian group that is distinct from cyclic group. When there is only one group of order k, then k is in A003277 and this group is the cyclic group C_k.
Except for a(1) = 4, all the terms are odd, because of the existence of a non-abelian dihedral group D_{2*n} of order 2*n for each n > 2.
Every p^2, p prime, is a term and the 2 corresponding abelian groups are C_{p^2} and C_p X C_p.

			4 is a term because the 2 groups of order 4 that are C_4 and C_2 X C_2, the Klein four-group, are both abelian and a(1) = 4 because there is no smallest order with 2 abelian groups.
45 is a term because the 2 groups of order 45 that are C_45 and C_5  X C_3 X C_3 are both abelian.
99 is another term because the 2 groups of order 99 that are C_99 and C_11 X C_3 X C_3 are both abelian.

Mathematics Stack Exchange, Group of order 45 is abelian.

Equals A051532 \ A003277.
Cf. A000001, A000688.
A001248 is a subsequence.

f[p_, e_] := Product[1 - p^i, {i, 1, e}]; q[n_] := !CoprimeQ[EulerPhi[n], n] && Module[{fct = FactorInteger[n], e}, e = fct[[;; , 2]]; Max[e] < 3 && CoprimeQ[Abs[Times @@ f @@@ fct], n]]; Select[Range[1500], q] (* Amiram Eldar, Dec 18 2021 *)

m such that A000001(m) = A000688(m) > 1.

More terms from Michel Marcus, Dec 18 2021

A350342 Numbers k such that k^2 is an abelian order.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 97, 101, 103, 107, 109, 113, 115, 119, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 173, 179, 181, 185, 187, 191, 193, 197, 199, 209

Offset: 1

Jianing Song, Dec 25 2021

nonn

Such k must be squarefree. Actually, such k must be a cyclic number (A003277).
Number of the form p_1*p_2*...*p_r 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 A350340.
From the term 5 on, no term can be divisible by 2 or 3.

			For primes p, p 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*q 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), A350343, A350340.

A350344 is the subsequence 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
isA350342(n) = isA051532(n^2)

A350343(n) = a(n)^2.

A350344 Composite k such that k^2 is an abelian order.

Original entry on oeis.org

35, 65, 77, 85, 115, 119, 133, 143, 161, 185, 187, 209, 215, 217, 221, 235, 247, 259, 265, 299, 319, 323, 329, 335, 341, 365, 371, 377, 391, 403, 407, 413, 415, 427, 437, 451, 469, 481, 485, 493, 511, 515, 517, 527, 533, 535, 551, 553, 559, 565, 583, 589, 595

Offset: 1

Jianing Song, Dec 25 2021

nonn

Numbers k such that k^2 is an abelian order with at least 4 groups.
Number of the form p_1*p_2*...*p_r where r > 1, the p_i are distinct primes and no (p_j)^2-1 is divisible by any p_i.
The smallest number k such that k^2 is an abelian order with at least 8 groups is A350340(3) = 595.
No term can be divisible by 2 or 3.

			For primes p, q, if p^2 !== 1 (mod q), q^2 !== 1 (mod p), then p*q 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}.
95 is not a term since 95^2 = 5^2 * 19^2 is not an abelian order. Note that 95 itself is a cyclic number.

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

Cf. A051532 (abelian orders), A050384, A350340.

Equals A350342 \ ({1} U A000040).

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
isA350344(n) = (n>1) && !isprime(n) && isA051532(n^2)

A350345(n) = a(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

Jianing Song, Dec 25 2021

nonn

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.

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

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)

a(n) = A350344(n)^2.

A350421 Numbers p^2*q, p > q odd primes such that q does not divide p-1, and q does not divide p+1.

Original entry on oeis.org

245, 845, 847, 1445, 1859, 2023, 2527, 2645, 3179, 3703, 3757, 3971, 4693, 6137, 6727, 6845, 6877, 8993, 9245, 9251, 9583, 10051, 10571, 10933, 11045, 12493, 14045, 14297, 15059, 15463, 15979, 16337, 17797, 18259, 18491, 19343, 19663, 21853, 22103, 22445, 23273

Offset: 1

Bernard Schott, Dec 30 2021

nonn

As odd prime q does not divide p-1 and does not divide also p+1, then q >= 5, so p >= 7.
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.

			245 = 7^2 * 5, 5 and 7 are odd primes, 5 does not divide 7-1 = 10 and does not divide 7+1 = 8, hence 245 is a term.

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

Equals A350422 \ A350332.
Subsequence of A051532, A054395, A054753, A060687 and A350322.
Other subsequences of A054753 linked with order of groups: A079704, A143928, A349495, A350115, A350245.

f:=Factorisation; [n:n in [3..24000 ]|#PrimeDivisors(n) eq 2 and  f(n)[1][1] lt f(n)[2][1] and f(n)[1][2] eq 1 and f(n)[2][2] eq 2  and (f(n)[2][1]-1) mod f(n)[1][1] ne 0 and (f(n)[2][1]+1) mod f(n)[1][1] ne 0]; // Marius A. Burtea, Dec 30 2021

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]]]]; Select[Range[1, 24000, 2], q] (* Amiram Eldar, Dec 30 2021 *)

isok(m) = my(f=factor(m)); if (f[, 2] == [1, 2]~, my(p=f[2, 1], q=f[1, 1]); ((p-1) % q) && ((p+1) % q)); \\ Michel Marcus, Dec 30 2021

from sympy import integer_nthroot, primerange
def aupto(limit):
    aset, maxp = set(), integer_nthroot(limit**2, 3)[0]
    for p in primerange(3, maxp+1):
        pp = p*p
        for q in primerange(1, min(p, limit//pp+1)):
            if (p-1)%q != 0 and (p+1)%q != 0:
                aset.add(pp*q)
    return sorted(aset)
print(aupto(24000)) # Michael S. Branicky, Dec 30 2021

More terms from Marius A. Burtea and Hugo Pfoertner, Dec 30 2021

A060653 Minimal number of conjugacy classes (which is also the number of irreducible representations) in G where G is a finite group of order n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 5, 9, 4, 11, 4, 13, 5, 15, 7, 17, 6, 19, 5, 5, 7, 23, 5, 25, 8, 11, 10, 29, 9, 31, 11, 33, 10, 35, 6, 37, 11, 7, 10, 41, 7, 43, 14, 45, 13, 47, 8, 49, 14, 51, 7, 53, 10, 7, 8, 9, 16, 59, 5, 61, 17, 15, 13, 65, 18, 67, 8, 69, 19, 71, 6, 73

Offset: 1

Noam Katz (noamkj(AT)hotmail.com), Apr 17 2001

nonn

a(n) <= n with equality iff n belongs to sequence A051532.

			a(6) = 3 because there are two groups of order 6, the cyclic group with 6 classes and S_3 with 3 classes.

Eric M. Schmidt, Table of n, a(n) for n = 1..1023

A051532.

A060653 := function(n) local min, i; min := n; for i in [1..NumberSmallGroups(n)] do min := Minimum(min, NrConjugacyClasses(SmallGroup(n,i))); od; return min; end; # Eric M. Schmidt, Aug 30 2012

More terms from Eric M. Schmidt, Aug 30 2012

A061064 Maximal number of zeros in the character table of a group with n elements.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 3, 0, 2, 0, 4, 0, 3, 0, 12, 0, 9, 0, 8, 4, 5, 0, 27, 0, 6, 16, 12, 0, 25, 0, 48, 0, 8, 0, 36, 0, 9, 8, 75, 0, 49, 0, 20, 0, 11, 0, 108, 0, 50, 0, 24, 0, 81, 8, 147, 12, 14, 0, 100, 0, 15, 36, 192, 0, 121, 0, 32, 0, 98, 0, 243, 0, 18, 16

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 05 2001

A finite non-Abelian group G has an irreducible representation of degree >= 2 and the character of such representation always has a zero; so a(n) = 0 iff every group of order n is Abelian, i.e. n belongs to A051532.

			a(6) = 1 because the character table of the symmetric group S_3 is / 1, 1, 1 / 1, 1, -1 / 2, -1, 0 /.

Eric M. Schmidt, Table of n, a(n) for n = 1..1023

Cf. A051532.

A061064 := function(n) local max, i; max := 0; for i in [1..NumberSmallGroups(n)] do max := Maximum(max, Sum(Irr(SmallGroup(n,i)), chi->Number(chi, x->x=0))); od; return max; end; # Eric M. Schmidt, Aug 24 2012

Added terms a(n) for n>=24, Eric M. Schmidt, Aug 24 2012.

A241276 Number of partitions of n that come from sizes of conjugacy classes of groups of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 3, 2, 2, 1, 7, 1, 2, 2, 2, 1, 4, 1, 6, 1, 2, 1, 6, 1, 2, 2, 5, 1, 6, 1, 2, 1, 2, 1, 13, 1, 3, 1, 3, 1, 7, 2, 5, 2, 2, 1, 9, 1, 2, 2, 16, 1, 4, 1, 3, 1, 4, 1, 17, 1, 2, 2, 2, 1, 6, 1, 11, 3, 2, 1, 9, 1, 2, 1, 4, 1, 6, 1, 2, 2, 2, 1, 30, 1, 3, 1, 7

Offset: 1

W. Edwin Clark, Apr 18 2014

a(n) = 1 if every group of order n is abelian, that is, if n is in A051532.
Upper bounds are given by A000001 (number of groups of order n) and A018818 (number of partitions of n into divisors of n).
A077191 is an upper bound. - Eric M. Schmidt, Oct 16 2014

			If n = 6 there are two groups of order 6: Z_6, all of whose conjugacy classes are of order 1 giving the partition [1,1,1,1,1,1] and S_6, which has three conjugacy classes whose sizes are 1, 2 and 3, giving the partition [1,2,3]. Hence a(6) = 2.

Eric M. Schmidt, Table of n, a(n) for n = 1..1023
Wikipedia, Conjugacy Class

a:=[];;
for n in [1..100] do
  P:=[];
  for i in [1..NumberSmallGroups(n)] do
   g:=SmallGroup(n,i);
   cc:=ConjugacyClasses(g);
   L:=List(cc,Size);
   Sort(L);
   Add(P,L);
   P:=Set(P);
  od;
  Add(a,Length(P));
od;
a;

a := function(n) local i, p, P; P := []; for i in [1..NrSmallGroups(n)] do p := List(ConjugacyClasses(SmallGroup(n,i)), Size); Sort(p); MakeImmutable(p); AddSet(P, p); od; return Length(P); end; # Eric M. Schmidt, Oct 16 2014

A060938 Maximal degree of an irreducible representation of a group with n elements.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 4, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 3, 4, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 6, 5, 7, 3, 2, 1, 5, 1, 2, 3, 4, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 3, 2, 1, 6, 1, 5, 3, 2, 1, 6, 1, 2, 1

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001

nonn

a(n) = 1 iff every group of order n is Abelian i.e. n belongs to sequence A051532.

a(m)a(n) <= a(mn). - Eric M. Schmidt, Oct 17 2012

			a(6) = 2 because for the Abelian group with 6 elements the degrees are all 1 and for the symmetric group S_3 the degrees are 1,1,2.

Eric M. Schmidt, Table of n, a(n) for n = 1..1535

Cf. A051532.

A060938 := function(n) local max, divs, maxpos, degs, i; if (n=1) then return 1; fi; divs := DivisorsInt(n); maxpos := divs[Int(Length(divs)/2)]; max := 1; for i in [1..NumberSmallGroups(n)] do degs := CharacterDegrees(SmallGroup(n, i)); max := Maximum(max, degs[Length(degs)][1]); if (max = maxpos) then return max; fi; od; return max; end;

More terms from Eric M. Schmidt, Oct 17 2012

A243592 Numbers n such that there is no indecomposable group of order n.

Original entry on oeis.org

1, 15, 33, 35, 45, 51, 65, 69, 77, 85, 87, 91, 95, 99, 105, 115, 119, 123, 133, 135, 141, 143, 145, 153, 159, 161, 165, 175, 177, 185, 187, 195, 207, 209, 213, 215, 217, 221, 231, 235, 245, 247, 249, 255, 259, 261, 265, 267, 285, 287, 295, 297, 299, 303, 315, 319, 321, 323, 325, 329, 335, 339, 341, 345, 357, 365, 369, 371

Offset: 1

Joerg Arndt, Jun 07 2014

nonn

Numbers n such that A090751(n) = 0.

Includes all non-prime-power members of A051532. - Eric M. Schmidt, Jun 07 2014

Joerg Arndt, Table of n, a(n) for n = 1..506
Paul Erdős, Péter P. Pálfy, On the orders of directly indecomposable groups. Paul Erdős memorial collection. Discrete Math. 200 (1999), no. 1-3, 165-179.

cp's OEIS Frontend

A350152 Abelian orders m for which there exist at least 2 groups with order m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A350342 Numbers k such that k^2 is an abelian order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A350344 Composite k such that k^2 is an abelian order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A350345 Squares of composite numbers k that are abelian orders.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A350421 Numbers p^2*q, p > q odd primes such that q does not divide p-1, and q does not divide p+1.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Extensions

A060653 Minimal number of conjugacy classes (which is also the number of irreducible representations) in G where G is a finite group of order n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Extensions

A061064 Maximal number of zeros in the character table of a group with n elements.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs