A350115 -id:A350115 - cp's OEIS frontend

A000001 Number of groups of order n.

0, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 13, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, 1, 2, 3, 4, 1, 6, 1, 52, 15, 2, 1, 15, 1, 2, 1, 12, 1, 10, 1, 4, 2

Offset: 0

N. J. A. Sloane

Also, number of nonisomorphic subgroups of order n in symmetric group S_n. - Lekraj Beedassy, Dec 16 2004
Also, number of nonisomorphic primitives (antiderivatives) of the combinatorial species Lin[n-1], or X^{n-1}; see Rajan, Summary item (i). - Nicolae Boicu, Apr 29 2011
In (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008), a(n) is called the "group number of n", denoted by gnu(n), and the first occurrence of k is called the "minimal order attaining k", denoted by moa(k) (see A046057). - Daniel Forgues, Feb 15 2017
It is conjectured in (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008) that the sequence n -> a(n) -> a(a(n)) = a^2(n) -> a(a(a(n))) = a^3(n) -> ... -> consists ultimately of 1s, where a(n), denoted by gnu(n), is called the "group number of n". - Muniru A Asiru, Nov 19 2017
MacHale (2020) shows that there are infinitely many values of n for which there are more groups than rings of that order (cf. A027623). He gives n = 36355 as an example. It would be nice to have enough values of n to create an OEIS entry for them. - N. J. A. Sloane, Jan 02 2021
I conjecture that a(i) * a(j) <= a(i*j) for all nonnegative integers i and j. - Jorge R. F. F. Lopes, Apr 21 2024

			Groups of orders 1 through 10 (C_n = cyclic, D_n = dihedral of order n, Q_8 = quaternion, S_n = symmetric):
1: C_1
2: C_2
3: C_3
4: C_4, C_2 X C_2
5: C_5
6: C_6, S_3=D_6
7: C_7
8: C_8, C_4 X C_2, C_2 X C_2 X C_2, D_8, Q_8
9: C_9, C_3 X C_3
10: C_10, D_10

S. R. Blackburn, P. M. Neumann, and G. Venkataraman, Enumeration of Finite Groups, Cambridge, 2007.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 302, #35.
J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 209.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 150.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, A Foundation for Computer Science, Addison-Wesley Publ. Co., Reading, MA, 1989, Section 6.6 'Fibonacci Numbers' pp. 281-283.
M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964.
D. Joyner, 'Adventures in Group Theory', Johns Hopkins Press. Pp. 169-172 has table of groups of orders < 26.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.24, p. 481.
M. F. Newman and E. A. O'Brien, A CAYLEY library for the groups of order dividing 128. Group theory (Singapore, 1987), 437-442, de Gruyter, Berlin-New York, 1989.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. U. Besche and Ivan Panchenko, Table of n, a(n) for n = 0..2047 [Terms 1 through 2015 copied from Small Groups Library mentioned below. Terms 2016 - 2047 added by _Ivan Panchenko_, Aug 29 2009. 0 prepended by _Ray Chandler_, Sep 16 2015. a(1024) corrected by _Benjamin Przybocki_, Jan 06 2022]
H. A. Bender, A determination of the groups of order p^5, Ann. of Math. (2) 29, pp. 61-72 (1927).
Hans Ulrich Besche and Bettina Eick, Construction of finite groups, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.
Hans Ulrich Besche and Bettina Eick, The groups of order at most 1000 except 512 and 768, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 405-413.
H. U. Besche, B. Eick and E. A. O'Brien, The groups of order at most 2000, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1-4.
H. U. Besche, B. Eick, E. A. O'Brien and Max Horn, The Small Groups Library
H. U. Besche, B. Eick and E. A. O'Brien, Number of isomorphism types of finite groups of given order [gives incorrect a(1024)]
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.
Henry Bottomley, Illustration of initial terms
David Burrell, On the number of groups of order 1024, Communications in Algebra, 2021, 1-3.
J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, Math. Intell., Vol. 30, No. 2, Spring 2008.
Marek Dančo, Mikoláš Janota, Michael Codish, and João Jorge Araújo, Complete Symmetry Breaking for Finite Models, arXiv:2502.10155 [cs.LO], 2025. See p. 7. Mentions this sequence.
Yang-Hui He and Minhyong Kim, Learning Algebraic Structures: Preliminary Investigations, arXiv:1905.02263 [cs.LG], 2019.
Otto Hölder, Die Gruppen der Ordnungen p^3, pq^2, pqr, p^4, Math. Ann. 43 pp. 301-412 (1893).
Max Horn, Numbers of isomorphism types of finite groups of given order
Rodney James, The groups of order p^6 (p an odd prime), Math. Comp. 34 (1980), 613-637.
Rodney James and John Cannon, Computation of isomorphism classes of p-groups, Mathematics of Computation 23.105 (1969): 135-140.
Olexandr Konovalov, Crowdsourcing project for the database of numbers of isomorphism types of finite groups, Github (a list of gnu(n) for many n < 50000).
Desmond MacHale, Are There More Finite Rings than Finite Groups?, Amer. Math. Monthly (2020) Vol. 127, Issue 10, 936-938.
Mehdi Makhul, Josef Schicho, and Audie Warren, On Galois groups of type-1 minimally rigid graphs, arXiv:2306.04392 [math.CO], 2023.
G. A. Miller, Determination of all the groups of order 64, Amer. J. Math., 52 (1930), 617-634.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
Dayanand S. Rajan, The equations D^kY=X^n in combinatorial species, Discrete Mathematics 118 (1993) 197-206 North-Holland.
E. Rodemich, The groups of order 128, J. Algebra 67 (1980), no. 1, 129-142.
Gordon Royle, Combinatorial Catalogues. See subpage "Generators of small groups" for explicit generators for most groups of even order < 1000. [broken link]
D. Rusin, Asymptotics [Cached copy of lost web page]
Eric Weisstein's World of Mathematics, Finite Group
Wikipedia, Finite group
M. Wild, The groups of order sixteen made easy, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.
Gang Xiao, SmallGroup
Index entries for sequences related to groups
Index entries for "core" sequences

The main sequences concerned with group theory are A000001 (this one), A000679, A001034, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A000688, A060689, A051532.
Cf. A046058, A046059, A023675, A023676.
A003277 gives n for which A000001(n) = 1, A063756 (partial sums).
A046057 gives first occurrence of each k.
A027623 gives the number of rings of order n.

A000001 := Concatenation([0], List([1..500], n -> NumberSmallGroups(n))); # Muniru A Asiru, Oct 15 2017

D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n) : n in [1..1000] ]; // John Cannon, Dec 23 2006

GroupTheory:-NumGroups(n); # with(GroupTheory); loads this command - N. J. A. Sloane, Dec 28 2017

FiniteGroupCount[Range[100]] (* Harvey P. Dale, Jan 29 2013 *)
a[ n_] := If[ n < 1, 0, FiniteGroupCount @ n]; (* Michael Somos, May 28 2014 *)

From Mitch Harris, Oct 25 2006: (Start)
For p, q, r primes:
a(p) = 1, a(p^2) = 2, a(p^3) = 5, a(p^4) = 14, if p = 2, otherwise 15.
a(p^5) = 61 + 2*p + 2*gcd(p-1,3) + gcd(p-1,4), p >= 5, a(2^5)=51, a(3^5)=67.
a(p^e) ~ p^((2/27)e^3 + O(e^(8/3))).
a(p*q) = 1 if gcd(p,q-1) = 1, 2 if gcd(p,q-1) = p. (p < q)
a(p*q^2) is one of the following:
   ---------------------------------------------------------------------------
  | a(p*q^2) |            p*q^2 of the form             |  Sequences (p*q^2)  |
   ---------- ------------------------------------------ ---------------------
  | (p+9)/2  |          q == 1 (mod p), p odd           |      A350638        |
  |    5     |         p=3, q=2 =>  p*q^2 = 12          |Special case with A_4|
  |    5     |               p=2, q odd                 |      A143928        |
  |    5     |             p == 1 (mod q^2)             |      A350115        |
  |    4     | p == 1 (mod q), p > 3, p !== 1 (mod q^2) |      A349495        |
  |    3     |       q == -1 (mod p), p and q odd       |      A350245        |
  |    2     |  q !== +-1 (mod p) and p !== 1 (mod q)   |      A350422        |
   ---------------------------------------------------------------------------
[Table from Bernard Schott, Jan 18 2022]
a(p*q*r) (p < q < r) is one of the following:
  q == 1 (mod p)  r == 1 (mod p)  r == 1 (mod q)  a(p*q*r)
  --------------  --------------  --------------  --------
        No              No              No            1
        No              No              Yes           2
        No              Yes             No            2
        No              Yes             Yes           4
        Yes             No              No            2
        Yes             No              Yes           3
        Yes             Yes             No           p+2
        Yes             Yes             Yes          p+4
[table from Derek Holt].
(End)
a(n) = A000688(n) + A060689(n). - R. J. Mathar, Mar 14 2015

More terms from Michael Somos

Typo in b-file description fixed by David Applegate, Sep 05 2009

A054753 Numbers which are the product of a prime and the square of a different prime (p^2 * q).

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428, 436, 452

Offset: 1

Henry Bottomley, Apr 25 2000

nonn

A178254(a(n)) = 4; union of A095990 and A096156. - Reinhard Zumkeller, May 24 2010
Numbers with prime signature (2,1) = union of numbers with ordered prime signature (1,2) and numbers with ordered prime signature (2,1) (restating second part of above comment). - Daniel Forgues, Feb 05 2011
A056595(a(n)) = 4. - Reinhard Zumkeller, Aug 15 2011
For k>1, Sum_{n>=1} 1/a(n)^k = P(k) * P(2*k) - P(3*k), where P is the prime zeta function. - Enrique Pérez Herrero, Jun 27 2012
Also numbers n with A001222(n)=3 and A001221(n)=2. - Enrique Pérez Herrero, Jun 27 2012
A089233(a(n)) = 2. - Reinhard Zumkeller, Sep 04 2013
Subsequence of the triprimes (A014612). If a(n) is even, then a(n)/2 is semiprime (A001358). - Wesley Ivan Hurt, Sep 08 2013
From Bernard Schott, Sep 16 2017: (Start)
These numbers are called "Nombres d'Einstein" on the French site "Diophante" (see link) because a(n) = m * c^2 where m and c are two different primes.
The numbers 44 = 2^2 * 11 and 45 = 3^2 * 5 are the two smallest consecutive "Einstein numbers"; 603, 604, 605 are the three smallest consecutive integers in this sequence. It's not possible to get more than five such consecutive numbers (proof in the link); the first set of five such consecutive numbers begins at the 17-digit number 10093613546512321. Where does the first sequence of four consecutive "Einstein numbers" begin? (End) [corrected by Jon E. Schoenfield, Sep 20 2017]
The first set of four consecutive integers in this sequence begins at the 11-digit number 17042641441. (Each such set must include two even numbers, one of which is of the form 2^2*q, the other of the form p^2*2; a quick search, taking the factorizations of consecutive integers before and after numbers of the latter form, shows that the number of sets of four consecutive k-digit integers in this sequence is 1, 7, 12, 18 for k = 11, 12, 13, 14, respectively.) - Jon E. Schoenfield, Sep 16 2017
The first 13 sets of 5 consecutive integers in this sequence have as their first terms 10093613546512321, 14414905793929921, 266667848769941521, 562672865058083521, 1579571757660876721, 1841337567664174321, 2737837351207392721, 4456162869973433521, 4683238426747860721, 4993613853242910721, 5037980611623036721, 5174116847290255921, 5344962129269790721. Each of these numbers except for the last is 7^2 times a prime; the last is 23^2 times a prime. - Jon E. Schoenfield, Sep 17 2017

			a(1) = 12 because 12 = 2^2*3 is the smallest number of the form p^2*q.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Guilhem Castagnos, Antoine Joux, Fabien Laguillaumie, and Phong Q. Nguyen, Factoring pq^2 with quadratic forms: nice cryptanalyses, Advances in Cryptology - ASIACRYPT 2009. Lecture Notes in Computer Science Volume 5912 (2009), pp. 469-486.
Diophante, A 350, Les Nombres d'Einstein (in French).
Mathematics Stack Exchange, Sequence of numbers with prime factorization pq^2
René Peralta and Eiji Okamoto, Faster factoring of integers of a special form (1996).
Index to sequences related to prime signature

Cf. A001221, A001222, A001358, A014612, A056595, A089233, A095990, A096156, A178254.
Numbers with 6 divisors (A030515) which are not 5th powers of primes (A050997).
Subsequence of A325241.  Supersequence of A096156.
Table giving for each subsequence the corresponding number of groups of order p^2*q, from Bernard Schott, Jan 23 2022
  -------------------------------------------------------------------------------
  | Subsequence  | A350638 | A143928 | A350115 | A349495 | A350245 | A350422 (*)|
  -------------------------------------------------------------------------------
  |A000001(p^2*q)| (q+9)/2 |    5    |    5    |    4    |    3    |      2     |
  -------------------------------------------------------------------------------
(*) A350422 equals disjoint union of A350332 (pA350421 (p>q).

Select[Range[12,452], {1,2}==Sort[Last/@FactorInteger[ # ]]&] (* Zak Seidov, Jul 19 2009 *)
With[{nn=60},Take[Union[Flatten[{#[[1]]#[[2]]^2,#[[1]]^2 #[[2]]}&/@ Subsets[ Prime[Range[nn]],{2}]]],nn]] (* Harvey P. Dale, Dec 15 2014 *)

is(n)=vecsort(factor(n)[,2])==[1,2]~ \\ Charles R Greathouse IV, Dec 30 2014

for(n=1, 1e3, if(numdiv(n) - bigomega(n) == 3, print1(n, ", "))) \\ Altug Alkan, Nov 24 2015

from sympy import factorint
def ok(n): return sorted(factorint(n).values()) == [1, 2]
print([k for k in range(453) if ok(k)]) # Michael S. Branicky, Dec 18 2021

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A054753(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p**2) for p in primerange(isqrt(x)+1))+primepi(integer_nthroot(x,3)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Link added and incorrect Mathematica code removed by David Bevan, Sep 17 2011

A350245 Numbers p^2*q, p > q odd primes such that q divides p+1.

Original entry on oeis.org

75, 363, 867, 1183, 1587, 1805, 2523, 4205, 5043, 6627, 8427, 10443, 11767, 15123, 17405, 20339, 20667, 23763, 26011, 30603, 31205, 34347, 38307, 39605, 48223, 51483, 56307, 59405, 65863, 66603, 76313, 83667, 89787, 96123, 96605, 109443, 111005, 115351, 116427

Offset: 1

Bernard Schott, Dec 21 2021

nonn

For these terms m, there are precisely 3 groups of order m, so this is a subsequence of A055561.

The 3 groups are C_{p^2*q}, (C_p X C_p) X C_q and (C_p X C_p) : C_q, where C means cyclic groups of the stated order, the symbols X and : mean direct and semidirect products respectively.

			75 = 5^2 * 3, 5 and 3 are odd and 3 divides 5+1 = 6, hence 75 is a term.
1183 = 13^2 * 7, 13 and 7 are odd and 7 divides 13+1 = 14, hence 1183 is another term.

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

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A054753 and A055561.

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

N:= 10^6: # for terms <= N
P:= select(isprime, [seq(i,i=3..floor(sqrt(N/3)),2)]):
g:= proc(p) local Q;
      Q:= numtheory:-factorset(p+1) minus {2};
      select(`<=`, map(q -> p^2*q, Q), N);
end proc:
sort(convert(`union`(op(map(g,P))),list)); # Robert Israel, Dec 28 2021

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; e == {1, 2} && Divisible[p[[2]] + 1, p[[1]]]]; Select[Range[1, 2*10^5, 2], q] (* Amiram Eldar, Dec 21 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(3, min(p-1, limit//pp)+1):
            if (p+1)%q == 0:
                aset.add(pp*q)
    return sorted(aset)
print(aupto(120000)) # Michael S. Branicky, Dec 21 2021

More terms from Amiram Eldar, Dec 21 2021

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

A350638 Numbers of the form p^2*q, with odd primes p > q, such that q divides p-1.

Original entry on oeis.org

147, 507, 605, 1083, 2883, 4107, 4805, 5547, 5819, 5887, 8405, 11163, 12943, 13467, 15987, 18605, 18723, 25205, 28227, 31827, 35287, 35643, 36517, 48387, 49379, 50807, 51005, 57963, 68403, 73947, 79707, 81133, 85805, 87131, 89383, 98283, 100949, 111747, 112903

Offset: 1

Bernard Schott, Jan 10 2022

nonn

For these terms m there are precisely (q+9)/2 groups of order m.

Only two of these groups are abelian: C_{p^2*q} and (C_p X C_p) X C_q. The (q+5)/2 groups that are nonabelian are C_{p^2} : C_q and the (q+3)/2 semidirect products of the form (C_p X C_p) : C_q that are not isomorphic, where C means cyclic groups of the stated order, the symbols X and : mean direct and semidirect products respectively.

			147 = 7^2 * 3, 3 and 7 are odd primes, 3 divides 7-1 = 6, hence 147 is a term.

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

N. S. Wedd, Groups of orders 147.

Other subsequences of A054753 linked with order of groups: A079704, A143928, A349495, A350115, A350245, A350332, A350421, A350422.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; e == {1, 2} && Divisible[p[[2]] - 1, p[[1]]]]; Select[Range[1, 120000, 2], q] (* Amiram Eldar, Jan 11 2022 *)

isok(m) = if (m%2, my(f=factor(m)); if (f[, 2] == [1, 2]~, my(p=f[1, 1], q=f[2, 1]); ((q-1) % p) == 0)); \\ Michel Marcus, Jan 11 2022

from sympy import integer_nthroot, primerange
def aupto(limit):
    aset, maxp = set(), integer_nthroot(limit**2, 3)[0]
    for p in primerange(5, maxp+1):
        pp = p*p
        for q in primerange(3, min(p, limit//pp+1)):
            if (p-1)%q == 0:
                aset.add(pp*q)
    return sorted(aset)
print(aupto(113000)) # Michael S. Branicky, Jan 10 2022

More terms from Michael S. Branicky, Jan 10 2022

cp's OEIS Frontend

A000001 Number of groups of order n.

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A054753 Numbers which are the product of a prime and the square of a different prime (p^2 * q).

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A350245 Numbers p^2*q, p > q odd primes such that q divides p+1.

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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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