A054395 -id:A054395 - 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

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

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

A215935 Number of ordered pairs of primes (p, q) dividing n for which p^e = 1 mod q, where e is the exponent of p in n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 1, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 1, 2, 1, 1, 0, 3, 0, 1, 1, 0, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 1, 1, 0, 3, 0, 2, 0, 1, 0, 4, 0, 1, 0, 1, 0, 2, 0, 1, 1

Offset: 1

Charles R Greathouse IV, Aug 27 2012

nonn

If n in A056867 then a(n) = 0.

			12 is divisible by two primes, 2 and 3. The exponent of 2 is 2 and the exponent of 3 is 1. 2^2 = 1 mod 3 and 3^1 = 1 mod 2, so a(12) = 2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A054395, A056867.

a:= proc(n) local l; l:= ifactors(n)[2];
       add(add(`if`(irem(i[1]^i[2], j[1])=1, 1, 0), i=l), j=l)
    end:
seq (a(n), n=1..100);  # Alois P. Heinz, Aug 28 2012

a[n_] := With[{f = FactorInteger[n]}, Sum[ Boole[ Mod[p[[1]]^p[[2]], q[[1]]] == 1], {p, f}, {q, f}]]; Table[a[n], {n, 1, 93}] (* Jean-François Alcover, Sep 03 2012 *)

a(n)=my(f=factor(n),k=#f~); sum(i=1,k, sum(j=1,k, i!=j && Mod(f[i,1],f[j,1])^f[i,2]==1))

A296026 Numbers n such that there are precisely 2 groups of order n and 4 of order n + 1.

Original entry on oeis.org

62, 129, 153, 169, 237, 245, 274, 278, 285, 289, 314, 369, 386, 417, 425, 429, 497, 529, 555, 597, 637, 645, 669, 715, 813, 889, 914, 926, 955, 961, 969, 1065, 1101, 1131, 1154, 1221, 1227, 1286, 1341, 1369, 1389, 1405, 1435, 1461, 1497, 1505, 1515, 1557, 1569, 1675, 1771

Offset: 1

Muniru A Asiru, Dec 03 2017

nonn

			62 is in the sequence since A000001(62) = 2 and A000001(63) = 4.
129 is in the sequence since A000001(129) = 2 and A000001(130) = 4.
425 is in the sequence since A000001(425) = 2 and A000001(426) = 4.

Muniru A Asiru, Table of n, a(n) for n = 1..1241
H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups

Cf. A000001. Subsequence of A054395.

A296026 := Filtered([1..2014],n->[NumberSmallGroups(n), NumberSmallGroups(n+1)]=[2, 4]);

with(GroupTheory):
for n from 1 to 10^3 do if [NumGroups(n), NumGroups(n+1)]=[2, 4]  then print(n); fi; od;

Sequence is { n | A000001(n) = 2, A000001(n+1) = 4 }.

A296025 Numbers n such that there are precisely 2 groups of order n and 3 of order n + 1.

Original entry on oeis.org

74, 362, 866, 2066, 2174, 3974, 4894, 5042, 5914, 6626, 7934, 10334, 10886, 12634, 15122, 16538, 17474, 19238, 20318, 20338, 20666, 21974, 23774, 23882, 24422, 25094, 28922, 31478, 33134, 35138, 36878, 38174, 41018, 41774, 42062, 42134, 46022, 48502

Offset: 1

Muniru A Asiru, Dec 03 2017

nonn

			74 is in the sequence since A000001(74) = 2 and A000001(75) = 3.
362 is in the sequence since A000001(362) = 2 and A000001(363) = 3.
7934 is in the sequence since A000001(7934) = 2 and A000001(7935) = 3.

H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups

Cf. A000001. Subsequence of A054395.

with(GroupTheory): with(numtheory):
for n from 1 to 10^4 do if [NumGroups(n), NumGroups(n+1)]=[2, 3] then print(n); fi; od;

Sequence is { n | A000001(n) = 2, A000001(n+1) = 3 }.

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cp's OEIS Frontend

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

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

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

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

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A215935 Number of ordered pairs of primes (p, q) dividing n for which p^e = 1 mod q, where e is the exponent of p in n.

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Maple

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A296026 Numbers n such that there are precisely 2 groups of order n and 4 of order n + 1.

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