A000001 -id:A000001 - cp's OEIS frontend

A298739 First differences of A000001 (the number of groups of order n).

0, 0, 1, -1, 1, -1, 4, -3, 0, -1, 4, -4, 1, -1, 13, -13, 4, -4, 4, -3, 0, -1, 14, -13, 0, 3, -1, -3, 3, -3, 50, -50, 1, -1, 13, -13, 1, 0, 12, -13, 5, -5, 3, -2, 0, -1, 51, -50, 3, -4, 4, -4, 14, -13, 11, -11, 0, -1, 12, -12, 1, 2, 263, -266, 3, -3

Offset: 1

Muniru A Asiru, Jan 25 2018

sign

			There is only one group of order 1 and of order 2, so a(1) = A000001(2) - A000001(1) = 1 - 1 = 0.
There are 2 groups of order 4 and 3 is a cyclic number, so a(3) = A000001(4) - A000001(3) = 2 - 1 = 1.

Muniru A Asiru, Table of n, a(n) for n = 1..2046 [a(1023) and a(1024) corrected by Andrey Zabolotskiy]
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 [dead link]
Index entries for sequences related to groups

Cf. A000001 (Number of groups of order n).

List([1..700],n -> NumberSmallGroups(n+1) - NumberSmallGroups(n));

with(GroupTheory): seq((NumGroups(n+1) - NumGroups(n), n=1..500));

(* Please note that, as of version 14, the Mathematica function FiniteGroupCount returns a wrong value for n = 1024 (49487365422 instead of 49487367289). *)
Differences[FiniteGroupCount[Range[100]]] (* Paolo Xausa, Mar 22 2024 *)

a(n) = A000001(n+1) - A000001(n).

A111101 Cumulative sum of A000001(n)^A000001(n).

Original entry on oeis.org

1, 2, 3, 7, 8, 12, 13, 3138, 3142, 3146, 3147, 6272, 6273, 6277, 6278, 11112006825564294, 11112006825564295, 11112006825567420, 11112006825567421, 11112006825570546, 11112006825570550, 11112006825570554, 11112006825570555, 449005897206429930, 449005897206429934

Offset: 1

Jonathan Vos Post, Oct 13 2005

Cf. A000001, A000312.

Accumulate[Table[FiniteGroupCount[n]^FiniteGroupCount[n],{n,25}]] (* James C. McMahon, Apr 30 2024 *)

a(n) = Sum_{i=1..n} A000001(i)^A000001(i).

a(n) = Sum_{i=1..n} A000312(A000001(i)).

More terms from James C. McMahon, Apr 30 2024

A142864 Value of A000001(n) as n runs through the triprimes (A014612).

Original entry on oeis.org

5, 5, 5, 5, 5, 4, 4, 6, 4, 2, 5, 5, 4, 4, 5, 4, 3, 4, 6, 4, 5, 2, 4, 2, 6, 6, 5, 4, 4, 5, 4, 4, 6, 5, 2, 4, 5, 2, 4, 5, 4, 4, 2, 4, 6, 4, 4, 2, 2, 5, 6, 4, 2, 4, 4, 5, 5, 2, 4, 1, 6, 2, 4, 4, 5, 4, 4, 4, 4, 2, 4, 4, 5, 6, 4, 4, 4, 2, 4, 5, 5, 5, 1, 4, 5, 2, 3, 6, 2, 4, 4, 2, 4, 5, 5, 6, 5, 6, 6, 4, 4, 2, 2, 4, 4

Offset: 1

N. J. A. Sloane, Oct 03 2008

nonn

John H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, The Mathematical Intelligencer, March 2008, Volume 30, Issue 2, pp 6-15.

More terms from R. J. Mathar, Oct 04 2008

A142865 Value of A000001(n) as n runs through the quadruprimes (A014613).

Original entry on oeis.org

14, 15, 14, 14, 15, 13, 13, 15, 15, 12, 10, 16, 14, 16, 10, 5, 15, 11, 13, 12, 18, 12, 13, 12, 10, 12, 12, 15, 6, 15, 14, 16, 12, 15, 15, 10, 23, 14, 5, 10, 9, 4, 15, 12, 15, 18, 12, 12, 10, 14, 11, 15, 7, 12, 11, 12, 10, 14, 13, 18, 5, 11, 12, 12, 11, 12, 14, 10, 12, 4, 8, 15, 15, 10

Offset: 1

N. J. A. Sloane, Oct 03 2008

nonn

John H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica.

More terms from R. J. Mathar, Oct 04 2008

A208663 Non-Abelian numbers: n such that A000001(n)/A000688(n) is a new record.

Original entry on oeis.org

1, 6, 12, 16, 24, 32, 48, 64, 96, 128, 256, 512, 1024, 2048

Offset: 1

Ben Branman, Feb 29 2012

			For a(n)=12, there are 2 Abelian groups and 3 nonabelian groups, so the ratio A000001(12)/A000688(12)=5/2=2.5, which beats the previous record of 2, so 12 is in the sequence.

H. A. Bender, A determination of the groups of order p^5, Ann. of Math. (2) 29, pp. 61-72 (1927).
H. U. Besche and B. Eick, Construction of Finite Groups, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.
H. U. Besche and B. 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, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.
M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964.
G. A. Miller, Determination of all the groups of order 64, Amer. J. Math., 52 (1930), 617-634.
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.
E. Rodemich, The groups of order 128. J. Algebra 67 (1980), no. 1, 129-142.

J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica.
Eric Weisstein's World of Mathematics, Finite Group
M. Wild, The groups of order sixteen made easy, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.
Index entries for sequences related to groups

Cf. A000001, A000688, A060689, A051532.

s = {1}; a = 1; Do[b = FiniteGroupCount[n]/FiniteAbelianGroupCount[n];
  If[b > a, a = b; AppendTo[s, n]], {n, 1, 2047}]; s

a(14) from Eric M. Schmidt, Aug 02 2012

A000688 Number of Abelian groups of order n; number of factorizations of n into prime powers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1

Offset: 1

N. J. A. Sloane

Equivalently, number of Abelian groups with n conjugacy classes. - Michael Somos, Aug 10 2010
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).
Also number of rings with n elements that are the direct product of fields; these are the commutative rings with n elements having no nilpotents; likewise the commutative rings where for every element x there is a k > 0 such that x^(k+1) = x. - Franklin T. Adams-Watters, Oct 20 2006
Range is A033637.
a(n) = 1 if and only if n is from A005117 (squarefree numbers). See the Ahmed Fares comment there, and the formula for n>=2 below. - Wolfdieter Lang, Sep 09 2012
Also, from a theorem of Molnár (see [Molnár]), the number of (non-isomorphic) abelian groups of order 2*n + 1 is equal to the number of non-congruent lattice Z-tilings of R^n by crosses, where a "cross" is a unit cube in R^n for which at each facet is attached another unit cube (Z, R are the integers and reals, respectively). (Cf. [Horak].) - L. Edson Jeffery, Nov 29 2012
Zeta(k*s) is the Dirichlet generating function of the characteristic function of numbers which are k-th powers (k=1 in A000012, k=2 in A010052, k=3 in A010057, see arXiv:1106.4038 Section 3.1). The infinite product over k (here) is the number of representations n=product_i (b_i)^(e_i) where all exponents e_i are distinct and >=1. Examples: a(n=4)=2: 4^1 = 2^2. a(n=8)=3: 8^1 = 2^1*2^2 = 2^3. a(n=9)=2: 9^1 = 3^2. a(n=12)=2: 12^1 = 3*2^2. a(n=16)=5: 16^1 = 2*2^3 = 4^2 = 2^2*4^1 = 2^4. If the e_i are the set {1,2} we get A046951, the number of representations as a product of a number and a square. - R. J. Mathar, Nov 05 2016
See A060689 for the number of non-abelian groups of order n. - M. F. Hasler, Oct 24 2017
Kendall & Rankin prove that the density of {n: a(n) = m} exists for each m. - Charles R Greathouse IV, Jul 14 2024

			a(1) = 1 since the trivial group {e} is the only group of order 1, and it is Abelian; alternatively, since the only factorization of 1 into prime powers is the empty product.
a(p) = 1 for any prime p, since the only factorization into prime powers is p = p^1, and (in view of Lagrange's theorem) there is only one group of prime order p; it is isomorphic to (Z/pZ,+) and thus Abelian.
From _Wolfdieter Lang_, Jul 22 2011: (Start)
a(8) = 3 because 8 = 2^3, hence a(8) = pa(3) = A000041(3) = 3 from the partitions (3), (2, 1) and (1, 1, 1), leading to the 3 factorizations of 8: 8, 4*2 and 2*2*2.
a(36) = 4 because 36 = 2^2*3^2, hence a(36) = pa(2)*pa(2) = 4 from the partitions (2) and (1, 1), leading to the 4 factorizations of 36: 2^2*3^2, 2^2*3^1*3^1, 2^1*2^1*3^2 and 2^1*2^1*3^1*3^1.
(End)

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 274-278.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.12, p. 468.
J. S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 7.
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).
A. Speiser, Die Theorie der Gruppen von endlicher Ordnung, 4. Auflage, Birkhäuser, 1956.

T. D. Noe, Table of n, a(n) for n = 1..10000
Tak-Shing T. Chan, and Y.-H. Yang, Polar n-Complex and n-Bicomplex Singular Value Decomposition and Principal Component Pursuit, IEEE Transactions on Signal Processing ( Volume: 64, Issue: 24, Dec.15, 15 2016 ); DOI: 10.1109/TSP.2016.2612171.
I. G. Connell, A number theory problem concerning finite groups and rings, Canad. Math. Bull, 7 (1964), 23-34.
I. G. Connell, Letter to N. J. A. Sloane, no date
P. Erdős and G. Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102.
Steven R. Finch, Abelian Group Enumeration Constants [Broken link]
Steven R. Finch, Abelian Group Enumeration Constants [broken link?] [From the Wayback machine]
P. Horak, Error-correcting codes and Minkowski's conjecture, Tatra Mt. Math. Publ., 45 (2010), p. 40.
B. Horvat, G. Jaklic and T. Pisanski, On the number of Hamiltonian groups, arXiv:math/0503183 [math.CO], 2005.
D. G. Kendall, R. A. Rankin, On the number of Abelian groups of a given order, Q. J. Math. 18 (1947) 197-208.
Nobushige Kurokawa and Masato Wakayama, Zeta extensions. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 7, 126--130. MR1930216 (2003h:11112).
E. Molnár, Sui mosaici dello spazio di dimensione n, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 51 (1971), 177-185.
H.-E. Richert, Über die Anzahl Abelscher Gruppen gegebener Ordnung I, Math. Zeitschr. 56 (1952) 21-32.
Marko Riedel, Counting Abelian Groups, Mathematics Stack Exchange, October 2014.
Laszlo Toth, A note on the number of abelian groups of a given order, arXiv:1203.6473 [math.NT], (2012).
Eric Weisstein's World of Mathematics, Abelian Group
Eric Weisstein's World of Mathematics, Finite Group
Eric Weisstein's World of Mathematics, Kronecker Decomposition Theorem
Index entries for sequences related to groups
Index entries for "core" sequences

Cf. A000001, A021002, A060689, A000041, A000961, A001055, A005361, A034382, A046054, A046055, A046056, A046101, A050360, A055653, A057521, A101872 (bisection), A101876 (quadrisection), A124010, A050361, A051532, A129667 (Dirichlet inverse), A188585.

Cf. A080729 (Dgf at s=2), A369634 (Dgf at s=3).

a000688 = product . map a000041 . a124010_row
-- Reinhard Zumkeller, Aug 28 2014

with(combinat): readlib(ifactors): for n from 1 to 120 do ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*numbpart(ifactors(n)[2][i][2]) od: printf(`%d,`,ans): od: # James Sellers, Dec 07 2000

f[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; Array[f, 107] (* Robert G. Wilson v, Sep 22 2006 *)
Table[FiniteAbelianGroupCount[n], {n, 200}] (* Requires version 7.0 or later. - Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)

A000688(n)=local(f);f=factor(n);prod(i=1,matsize(f)[1],numbpart(f[i,2])) \\ Michael B. Porter, Feb 08 2010

a(n)=my(f=factor(n)[,2]); prod(i=1,#f,numbpart(f[i])) \\ Charles R Greathouse IV, Apr 16 2015

from sympy import factorint, npartitions
from math import prod
def A000688(n): return prod(map(npartitions,factorint(n).values())) # Chai Wah Wu, Jan 14 2022

def a(n):
    F=factor(n)
    return prod([number_of_partitions(F[i][1]) for i in range(len(F))])
# Ralf Stephan, Jun 21 2014

Multiplicative with a(p^k) = number of partitions of k = A000041(k); a(mn) = a(m)a(n) if (m, n) = 1.
a(2n) = A101872(n).
a(n) = Product_{j = 1..N(n)} A000041(e(j)), n >= 2, if
  n  = Product_{j = 1..N(n)} prime(j)^e(j), N(n) = A001221(n). See the Richert reference, quoting A. Speiser's book on finite groups (in German, p. 51 in words). - Wolfdieter Lang, Jul 23 2011
In terms of the cycle index of the symmetric group: Product_{q=1..m} [z^{v_q}] Z(S_v) 1/(1-z) where v is the maximum exponent of any prime in the prime factorization of n, v_q are the exponents of the prime factors, and Z(S_v) is the cycle index of the symmetric group on v elements. - Marko Riedel, Oct 03 2014
Dirichlet g.f.: Sum_{n >= 1} a(n)/n^s = Product_{k >= 1} zeta(ks) [Kendall]. - Álvar Ibeas, Nov 05 2014
a(n)=2 for all n in A054753 and for all n in A085987.  a(n)=3 for all n in A030078 and for all n in A065036.  a(n)=4 for all n in A085986.  a(n)=5 for all n in A030514 and for all n in A178739.  a(n)=6 for all n in A143610. - R. J. Mathar, Nov 05 2016
A050360(n) = a(A025487(n)). a(n) = A050360(A101296(n)). - R. J. Mathar, May 26 2017
a(n) = A000001(n) - A060689(n). - M. F. Hasler, Oct 24 2017
From Amiram Eldar, Nov 01 2020: (Start)
a(n) = a(A057521(n)).
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = A021002. (End)
a(n) = A005361(n) except when n is a term of A046101, since A000041(x) = x for x <= 3. - Miles Englezou, Feb 17 2024
Inverse Moebius transform of A188585: a(n) = Sum_{d|n} A188585(d). - Amiram Eldar, Jun 10 2025

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

A005180 Orders of simple groups.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 60, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 168, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

N. J. A. Sloane, R. K. Guy

Officially the group of order 1 is not considered to be simple - see for example Rotman, Group Theory.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
M. Hall, Jr., A search for simple groups of order less than one million, pp. 137-168 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Aschbacher, Review of "Classification of the finite simple groups, No. 2" by D. Gorenstein, R. Lyons & R. Solomon
C. Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574-577.
Dept. of Pure Math., Univ. Sheffield, The Classification of Finite Simple Groups
D. Gorenstein, R. Lyons & R. Solomon, The Classification of the Finite Simple Groups, AMS Books Online, Providence RI 1994.
D. Gorenstein, R. Lyons & R. Solomon, The Classification of the Finite Simple Groups, Number 2, AMS Books Online, Providence RI 1996.
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
R. Solomon, A Brief History Of The Classification Of The Finite Simple Groups, Bull. Amer. Math. Soc. 38 (2001), 315-352.
Index entries for sequences related to groups

Cf. A000001, A001228. Union of {1}, A000040 and A001034.

(* Recomputation from A001034. *)
maxOrder = 7789;
A001034 = Select[Cases[Import["https://oeis.org/A001034/b001034.txt", "Table"], {, }][[All, 2]], # <= maxOrder&];
Union[{1}, Prime[Range[PrimePi[maxOrder]]], A001034] (* Jean-François Alcover, Aug 19 2019 *)

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

Original entry on oeis.org

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

A054397 Numbers m such that there are precisely 5 groups of order m.

Original entry on oeis.org

8, 12, 18, 20, 27, 50, 52, 68, 98, 116, 125, 135, 148, 164, 171, 212, 242, 244, 273, 292, 297, 333, 338, 343, 356, 388, 399, 404, 436, 452, 459, 548, 578, 596, 621, 628, 651, 657, 692, 722, 724, 741, 772, 777, 783, 788, 825, 855, 875, 916, 932, 964, 981

Offset: 1

N. J. A. Sloane, May 21 2000

nonn

For m = 2*p^2 (p prime), there are precisely 5 groups of order m, so A079704 and A143928 (p odd prime) are two subsequences. - Bernard Schott, Dec 10 2021
For m = p^3, p prime, there are also 5 groups of order m, so A030078, where these groups are described, is another subsequence. - Bernard Schott, Dec 11 2021
For m squarefree, there are 5 groups of order m if and only if all of the following hold: 3|m, there are exactly two prime factors p,q of m such that p,q = 1 mod 3, no other relations of the form p' = 1 mod q' hold for p',q' prime factors of m. - Robin Jones, May 27 2025

			For m = 8, the 5 groups of order 8 are C8, C4 x C2, D8, Q8, C2 x C2 x C2 and for m = 12 the 5 groups of order 12 are C3 : C4, C12, A4, D12, C6 x C2 where C, D, Q  mean cyclic, dihedral, quaternion groups of the stated order and A is the alternating group of the stated degree. The symbols x and : mean direct and semidirect products respectively. - _Muniru A Asiru_, Nov 03 2017

Jorge R. F. F. Lopes, Table of n, a(n) for n = 1..2035, (terms 1..120 from Muniru A Asiru and Georg Fischer).
H. U. Besche, B. Eick and E. A. O'Brien, The Small Groups Library
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), this sequence (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).

Cf. A384370 (squarefree numbers in this sequence).

A054397 := Filtered([1..2015], n -> NumberSmallGroups(n) = 5); # Muniru A Asiru, Nov 03 2017

Select[Range[10^4], FiniteGroupCount[#] == 5 &] (* Robert Price, May 23 2019 *)

Sequence is { k | A000001(k) = 5 }. - Muniru A Asiru, Nov 03 2017

More terms from Christian G. Bower, May 25 2000

cp's OEIS Frontend

A298739 First differences of A000001 (the number of groups of order n).

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A111101 Cumulative sum of A000001(n)^A000001(n).

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A142864 Value of A000001(n) as n runs through the triprimes (A014612).

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A142865 Value of A000001(n) as n runs through the quadruprimes (A014613).

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A208663 Non-Abelian numbers: n such that A000001(n)/A000688(n) is a new record.

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A000688 Number of Abelian groups of order n; number of factorizations of n into prime powers.

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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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A005180 Orders of simple groups.

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