A350322 -id:A350322 - cp's OEIS frontend

A054395 Numbers m such that there are precisely 2 groups of order m.

4, 6, 9, 10, 14, 21, 22, 25, 26, 34, 38, 39, 45, 46, 49, 55, 57, 58, 62, 74, 82, 86, 93, 94, 99, 105, 106, 111, 118, 121, 122, 129, 134, 142, 146, 153, 155, 158, 165, 166, 169, 175, 178, 183, 194, 195, 201, 202, 203, 205, 206, 207, 214, 218, 219, 226, 231, 237

Offset: 1

N. J. A. Sloane, May 21 2000

nonn

Givens characterizes this sequence, see Theorem 5. In particular, this sequence is ({n: A215935(n) = 1} INTERSECT A005117) UNION (A060687 INTERSECT A051532). - Charles R Greathouse IV, Aug 27 2012 [This is now A350586 UNION A350322. - Charles R Greathouse IV, Jan 08 2022]

Numbers m such that A000001(m) = 2. - Muniru A Asiru, Nov 03 2017

			For m = 4, the 2 groups of order 4 are C4, C2 x C2; for m = 6, the 2 groups of order 6 are S3, C6; and for m = 9, the 2 groups of order 9 are C9, C3 x C3 where C is the cyclic group of the stated order and S is the symmetric group of the stated degree. The symbol x means direct product. - _Muniru A Asiru_, Oct 24 2017

Muniru A Asiru and Gheorghe Coserea, Table of n, a(n) for n = 1..234567, terms 1..422 from Muniru A Asiru.
H. U. Besche, B. Eick and E. A. O'Brien, The Small Groups Library
Clint Givens, Orders for which there exist exactly two groups (2006)
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups

Equals A350586 UNION A350322.

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: this sequence (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

A054395 := Filtered([1..2015], n -> NumberSmallGroups(n) = 2); # Muniru A Asiru, Oct 24 2017

IsGivensInt := function(n)
  local p, f; p := GcdInt(n, Phi(n));
  if not IsPrimeInt(p) then return false; fi;
  if n mod p^2 = 0 then return 1 = GcdInt(p+1, n); fi;
  f := PrimePowersInt(n);
  return 1 = Number([1..QuoInt(Length(f),2)], k->f[2*k-1] mod p = 1);
end;;
Filtered([1..240], IsGivensInt); # Gheorghe Coserea, Dec 04 2017

Select[Range[240], FiniteGroupCount[#] == 2&]
(* or: *)
okQ[n_] := Module[{p, f}, p = GCD[n, EulerPhi[n]]; If[! PrimeQ[p], Return[False]]; If[Mod[n, p^2] == 0, Return[1 == GCD[p + 1, n]]]; f = FactorInteger[n]; 1 == Sum[Boole[Mod[f[[k, 1]], p] == 1], {k, 1, Length[f]}]];
Select[Range[240], okQ] (* Jean-François Alcover, Dec 08 2017, after Gheorghe Coserea *)

is(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);
};
seq(N) = {
  my(a = vector(N), k=0, n=1);
  while(k < N, if(is(n), a[k++]=n); n++); a;
};
seq(58) \\ Gheorghe Coserea, Dec 03 2017

More terms from Christian G. Bower, May 25 2000

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)

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

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

A350586 Numbers m with exactly 2 groups of order m, where one is abelian and the other is nonabelian.

Original entry on oeis.org

6, 10, 14, 21, 22, 26, 34, 38, 39, 46, 55, 57, 58, 62, 74, 82, 86, 93, 94, 105, 106, 111, 118, 122, 129, 134, 142, 146, 155, 158, 165, 166, 178, 183, 194, 195, 201, 202, 203, 205, 206, 214, 218, 219, 226, 231, 237, 253, 254, 262, 274, 278, 285, 291, 298, 301, 302

Offset: 1

Bernard Schott, Jan 07 2022

nonn

Differs from A064899 that is a subsequence: a(20) = 105 while A064899(20) = 106.
When m = 2*p, p odd prime, abelian group is C_{2*p} and nonabelian group is D_{2*p} ~ C_p : C_2.
When m = p*q, pIn both cases, C, D mean cyclic, dihedral groups of the stated order; the symbols ~ and : mean isomorphic to and semidirect product respectively.
A number m is a term iff m is squarefree and m has exactly one pair of prime factors (p, q) such that q == 1 (mod p). - David Radcliffe, Jul 30 2025

			There is only one group of order 1, 2, 3, 5 and the two groups of order 4 are abelian; hence 6 is the smallest term because the two groups of order 6 are the abelian and cyclic group C_6, while the nonabelian group is the symmetric group S_3 isomorphic to dihedral group D_6.
The smallest odd term is 21, the two corresponding groups are C_21 and semi-direct product C_7 : C_3.
The smallest term of the form p*q*r, p < q < r primes, is 105, the two corresponding groups are C_105 and semi-direct product C_35 : C_3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J.H. Conway, H. Dietrich, and E.A. O’Brien, Counting Groups: Gnus, Moas, and other Exotica, The Mathematical Intelligencer 30, 6-15 (2008), doi:10.1007/BF02985731.

Equals A054395 \ A350322.
Subsequence of A060650 and of A005117.
Cf. A000001, A064899.

is(n,f=factor(n))=my(p=f[,1],s); if(#p && vecmax(f[,2])>1, return(0)); for(i=2,#p, for(j=1,i-1, if(p[i]%p[j]==1 && s++>1, return(0)))); s==1 \\ Charles R Greathouse IV, Jan 08 2022

list(lim)=my(v=List()); forsquarefree(n=6,lim\1, my(p=n[2][,1],s); for(i=2,#p, for(j=1,i-1, if(p[i]%p[j]==1 && s++>1, next(3)))); if(s==1, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jan 08 2022

from sympy import factorint
def is_ok(m):
    f = factorint(m)
    if any(e > 1 for e in f.values()): return False # m must be squarefree
    return sum(q % p == 1 for p in f for q in f) == 1 # David Radcliffe, Jul 30 2025

More terms from Jinyuan Wang, Jan 08 2022

Showing 1-6 of 6 results.

cp's OEIS Frontend

A054395 Numbers m such that there are precisely 2 groups of order m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

GAP

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A350343 Square numbers k that are abelian orders.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

PARI

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

A350586 Numbers m with exactly 2 groups of order m, where one is abelian and the other is nonabelian.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Python

Extensions