A060652 -id:A060652 - cp's OEIS frontend

A386233 Number of good involutions of all nontrivial conjugation quandles of order A060652(n).

1, 32, 1, 17, 1, 13056, 66, 33, 1, 1

Offset: 1

Luc Ta, Jul 16 2025

A good involution f of a quandle Q is an involution that commutes with all inner automorphisms and satisfies the identity f(y)(x) = y^-1(x). We call the pair (Q,f) a symmetric quandle.

A conjugation quandle is a group viewed as a quandle under the conjugation operation. Since conjugation quandles of abelian groups are trivial, this sequence only considers nonabelian groups.

			For n = 1, 3, 5, 9, 10, there is a unique nonabelian group G of order A060652(n), and G is centerless. It follows from Ta, Prop. 5.3 that a(n) = 1.

Seiichi Kamada, Quandles with good involutions, their homologies and knot invariants, Intelligence of Low Dimensional Topology 2006, World Scientific Publishing Co. Pte. Ltd., 2007, 101-108.

Lực Ta, Good involutions of conjugation subquandles, arXiv:2505.08090 [math.GT], 2025. See Table 2.
Lực Ta, Symmetric-Rack-Classification, GitHub, 2025.

Cf. A000001, A060652, A060689, A386233, A181769, A383828, A386231, A386232.

Other sequences related to racks and quandles: A383144, A181771, A176077, A179010, A193024, A254434, A177886, A196111, A226173, A236146, A248908, A165200, A242044, A226193, A242275, A243931, A257351, A198147, A225744, A226172, A226174, A383828-A383831, A383145, A383146, A178432, A385041, A383145, A181770.

GAP
```
See Ta, GitHub link
```

A051532 The abelian orders (or abelian numbers): numbers m such that every group of order m is abelian.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 113, 115, 119, 121, 123, 127, 131, 133, 137, 139, 141, 143, 145, 149, 151, 153, 157, 159, 161

Offset: 1

Des MacHale, Dec 11 1999

Except for a(2)=2 and a(4)=4, all of the terms in the sequence are odd. This is because of the existence of a non-abelian dihedral group of order 2m for each m > 2.
Cubefree terms of A056867; A212793(a(n)) = 1. - Reinhard Zumkeller, Jun 28 2013
See similar comment with "squarefree terms" in A003277 (Donald J. McCarthy link). - Bernard Schott, Feb 20 2023

			a(4) = 4 because every group of order 4 is abelian.
These two abelian groups of order 4 are the cyclic group C_4 and the Klein four-group = C_2 X C_2, the smallest non-cyclic abelian group. - _Bernard Schott_, Feb 21 2023

W. R. Scott, Group Theory, Dover, 1987, page 217.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Donald J. McCarthy, A survey of partial converses to Lagrange's theorem on finite groups, Transactions of the New York Academy of Sciences, 1971, page 592.
J. Pakianathan and K. Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, August-September 2000, pp. 631-634.
Index entries for sequences related to groups

Subsequence of A056867 and supersequence of A003277.
Complement of A060652.
Intersection of A004709 and A056867.
Cf. A064899, A212793.

a051532 n = a051532_list !! (n-1)
a051532_list = filter ((== 1) . a212793) a056867_list
-- Reinhard Zumkeller, Jun 28 2013

okQ[n_] := Module[{f, lf, p, e, v}, f = FactorInteger[n]; lf = Length[f]; p = f[[All, 1]]; e = f[[All, 2]]; If[AnyTrue[e, # > 2&], Return[False]]; v = p^e; For[i = 1, i <= lf, i++, For[j = i+1, j <= lf, j++, If[Mod[v[[i]], p[[j]]]==1 || Mod[v[[j]], p[[i]]]==1, Return[False]]]]; Return[True]];
Select[Range[200], okQ] (* Jean-François Alcover, May 03 2012, after PARI, updated Jan 10 2020 *)

is(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, Feb 13 2011

from sympy import factorint
def ok(n):
    if n == 1: return True
    f = factorint(n)
    p, e = f.keys(), f.values()
    if max(e) >= 3: return False
    return all((pi**k)%pj!=1 for pi in p for pj in p if pj!=pi for k in range(1, f[pi]+1))
print([k for k in range(1, 162) if ok(k)]) # Michael S. Branicky, Feb 20 2023

m must be cubefree and its prime divisors must satisfy certain congruences.

Let the prime factorization of m be p1^e1 * ... * pr^er. Then m is in this sequence if ei < 3 for all i and pi^k is not congruent to 1 (mod pj) for all i and j and 1 <= k <= ei. - T. D. Noe, Mar 25 2007

A056866 Orders of non-solvable groups, i.e., numbers that are not solvable numbers.

Original entry on oeis.org

60, 120, 168, 180, 240, 300, 336, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 900, 960, 1008, 1020, 1080, 1092, 1140, 1176, 1200, 1260, 1320, 1344, 1380, 1440, 1500, 1512, 1560, 1620, 1680, 1740, 1800, 1848, 1860, 1920, 1980, 2016, 2040

Offset: 1

N. J. A. Sloane, Sep 02 2000

A number is solvable if every group of that order is solvable.
This comment is about the three sequences A001034, A060793, A056866: The Feit-Thompson theorem says that a finite group with odd order is solvable, hence all numbers in this sequence are even. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001 [Corrected by Isaac Saffold, Aug 09 2021]
Insoluble group orders can be derived from A001034 (simple non-cyclic orders): k is an insoluble order iff k is a multiple of a simple non-cyclic order. - Des MacHale
All terms are divisible by 4 and either 3 or 5. - Charles R Greathouse IV, Sep 11 2012
Subsequence of A056868 and hence of A060652. - Charles R Greathouse IV, Apr 16 2015, updated Sep 11 2015
The primitive elements are A257146. Since the sum of the reciprocals of the terms of that sequence converges, this sequence has a natural density and so a(n) ~ k*n for some k (see, e.g., Erdős 1948). - Charles R Greathouse IV, Apr 17 2015
From Jianing Song, Apr 04 2022: (Start)
Burnside's p^a*q^b theorem says that a finite group whose order has at most 2 distinct prime factors is solvable, hence all terms have at least 3 distinct prime factors.
Terms not divisible by 12 are divisible by 320 and have at least 4 distinct prime factors (cf. A257391). (End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2240 terms from T. D. Noe)
R. Brauer, Investigation on groups of even order, I.
R. Brauer, Investigation on groups of even order, II.
P. Erdős, On the density of some sequences of integers, Bull. Amer. Math. Soc. 54 (1948), pp. 685-692. See p. 685.
W. Feit and J. G. Thompson, A solvability criterion for finite groups and consequences, Proc. N. A. S. 48 (6) (1962) 968.
J. Pakianathan and K. Shankar, Nilpotent numbers, Amer. Math. Monthly, 107, August-September 2000, 631-634.
Cindy Tsang, Qin Chao, On the solvability of regular subgroups in the holomorph of a finite solvable group, arXiv:1901.10636 [math.GR], 2019.
Index entries for sequences related to groups

Subsequence of A000977 and A056868.

Cf. A003277, A051532, A056867, A001034, A060652.

ma[n_] := For[k = 1, True, k++, p = Prime[k]; m = 2^p*(2^(2*p) - 1); If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]; mb[n_] := For[k = 2, True, k++, p = Prime[k]; m = 3^p*((3^(2*p) - 1)/2); If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]; mc[n_] := For[k = 3, True, k++, p = Prime[k]; m = p*((p^2 - 1)/2); If[Mod[p^2 + 1, 5] == 0, If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]]; md[n_] := Mod[n, 2^4*3^3*13] == 0; me[n_] := For[k = 2, True, k++, p = Prime[k]; m = 2^(2*p)*(2^(2*p) + 1)*(2^p - 1); If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]; notSolvableQ[n_] := OddQ[n] || ma[n] || mb[n] || mc[n] || md[n] || me[n]; Select[ Range[3000], notSolvableQ] (* Jean-François Alcover, Jun 14 2012, from formula *)

is(n)={
    if(n%5616==0,return(1));
    forprime(p=2,valuation(n,2),
        if(n%(4^p-1)==0, return(1))
    );
    forprime(p=3,valuation(n,3),
        if(n%(9^p\2)==0, return(1))
    );
    forprime(p=3,valuation(n,2)\2,
        if(n%((4^p+1)*(2^p-1))==0, return(1))
    );
    my(f=factor(n)[,1]);
    for(i=1,#f,
        if(f[i]>3 && f[i]%5>1 && f[i]%5<4 && n%(f[i]^2\2)==0, return(1))
    );
    0
}; \\ Charles R Greathouse IV, Sep 11 2012

A positive integer k is a non-solvable number if and only if it is a multiple of any of the following numbers: a) 2^p*(2^(2*p)-1), p any prime. b) 3^p*(3^(2*p)-1)/2, p odd prime. c) p*(p^2-1)/2, p prime greater than 3 such that p^2 + 1 == 0 (mod 5). d) 2^4*3^3*13. e) 2^(2*p)*(2^(2*p)+1)*(2^p-1), p odd prime.

More terms from Des MacHale, Feb 19 2001

Further terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

A064899 Numbers that are of the form pq where p and q are distinct primes that are the orders of non-Abelian groups.

Original entry on oeis.org

6, 10, 14, 21, 22, 26, 34, 38, 39, 46, 55, 57, 58, 62, 74, 82, 86, 93, 94, 106, 111, 118, 122, 129, 134, 142, 146, 155, 158, 166, 178, 183, 194, 201, 202, 203, 205, 206, 214, 218, 219, 226, 237, 253, 254, 262, 274, 278, 291, 298, 301, 302, 305, 309, 314, 326

Offset: 1

Patrick De Geest, Oct 13 2001

nonn

This is the intersection of sequences A060652 and A006881.

Cf. A001358, A064900-A064911.

is(n)=my(f=factor(n)); f[,2]==[1,1]~ && f[2,1]%f[1,1]==1 \\ Charles R Greathouse IV, Oct 22 2015

list(lim)=my(v=List()); forprime(q=3,lim\2,forprime(p=2,min(lim\q,q-1), if(q%p==1, listput(v,p*q)))); Set(v) \\ Charles R Greathouse IV, Oct 22 2015

Definition corrected by Franklin T. Adams-Watters, Oct 09 2006

A056868 Numbers that are not nilpotent numbers.

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 21, 22, 24, 26, 28, 30, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 66, 68, 70, 72, 74, 75, 76, 78, 80, 82, 84, 86, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 117, 118, 120

Offset: 1

N. J. A. Sloane, Sep 02 2000

A number is nilpotent if every group of order n is nilpotent.

The sequence "Numbers of the form (k*i + 1)*k*j with i, j >= 1 and k >= 2" agrees with this for the first 146 terms but then differs. Cf. A300737. - Gionata Neri, Mar 11 2018

			From _Bernard Schott_, Dec 19 2021: (Start)
There are 2 groups with order 6: C_6 that is cyclic so nilpotent, and the symmetric group S_3 that is not nilpotent, hence 6 is a term.
There are also 2 groups with order 10: C_10 that is cyclic so nilpotent, and the dihedral group D_10 that is not nilpotent, hence 10 is another term. (End)

T. D. Noe, Table of n, a(n) for n = 1..10000
J. Pakianathan and K. Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, August-September 2000, pp. 631-634.

Complement of A056867.
Subsequence of A060652; A068919 is a subsequence.
Cf. A003277, A051532, A056866, A027748, A124010, A300737.

a056868 n = a056868_list !! (n-1)
a056868_list = filter (any (== 1) . pks) [1..] where
   pks x = [p ^ k `mod` q | let fs = a027748_row x, q <- fs,
                            (p,e) <- zip fs $ a124010_row x, k <- [1..e]]
-- Reinhard Zumkeller, Jun 28 2013

nilpotentQ[n_] := With[{f = FactorInteger[n]}, Sum[ Boole[ Mod[p[[1]]^p[[2]], q[[1]]] == 1], {p, f}, {q, f}]] == 0; Select[ Range[120], !nilpotentQ[#]& ] (* Jean-François Alcover, Sep 03 2012 *)

is(n)=my(f=factor(n));for(k=1,#f[,1], for(j=1,f[k,2], if(gcd(n, f[k,1]^j-1)>1, return(1)))); 0 \\ Charles R Greathouse IV, Sep 18 2012

n is in this sequence if p^k = 1 mod q for primes p and q dividing n such that p^k divides n. - Charles R Greathouse IV, Aug 27 2012

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

A068919 Squarefree orders of non-Abelian groups.

Original entry on oeis.org

6, 10, 14, 21, 22, 26, 30, 34, 38, 39, 42, 46, 55, 57, 58, 62, 66, 70, 74, 78, 82, 86, 93, 94, 102, 105, 106, 110, 111, 114, 118, 122, 129, 130, 134, 138, 142, 146, 154, 155, 158, 165, 166, 170, 174, 178, 182, 183, 186, 190, 194, 195, 201, 202, 203, 205, 206

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Mar 29 2002

nonn

This set is the difference A005117 \ A003277, that is, the squarefree numbers that are not cyclic (A003277). - Jeppe Stig Nielsen, Sep 25 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Intersection of A060652 and A005117.
Cf. A064899.
Also intersection of A056868 and A005117. - Reinhard Zumkeller, Jun 28 2013
Cf. A008966.

a068919 n = a068919_list !! (n-1)
a068919_list = filter ((== 1) . a008966) a056868_list
-- Reinhard Zumkeller, Jun 28 2013

Complement[Select[Range[206], SquareFreeQ[#] &],
Select[Range[206], FiniteGroupCount[#] == 1 &]] (* Geoffrey Critzer, Nov 28 2015 *)

is(n)=issquarefree(n)&&gcd(n,eulerphi(n))!=1 \\ Jeppe Stig Nielsen, Sep 25 2017

A221048 The odd semiprime numbers (A046315) which are orders of a non-Abelian group.

Original entry on oeis.org

21, 39, 55, 57, 93, 111, 129, 155, 183, 201, 203, 205, 219, 237, 253, 291, 301, 305, 309, 327, 355, 381, 417, 453, 471, 489, 497, 505, 543, 579, 597, 633, 655, 669, 687, 689, 723, 737, 755, 791, 813, 831, 849, 889, 905, 921, 939, 955, 979, 993, 1011, 1027, 1047

Offset: 1

David Brown, Apr 14 2013

nonn

Numbers of the form pq where p,q are odd primes, p

The corresponding non-Abelian groups are the semidirect products of Z/qZ and Z/pZ. - Bernard Schott, May 16 2020

Jinyuan Wang, Table of n, a(n) for n = 1..1000
S. K. Berberian, Non-Abelian Groups of Order pq, American Mathematical Monthly, Vol. 60, No. 1, Jan. 1953, 37-40.

Intersection of A046315 and A060652.

Select[1 + 2*Range[500], (f = FactorInteger[#]; Last /@ f == {1, 1} && Mod @@ Reverse[First /@ f] == 1) &] (* Giovanni Resta, Apr 14 2013 *)

lista(nn) = {forstep(n=1, nn, 2, my(f=factor(n)); if ((#f~ == 2) && (vecmax(f[,2]) == 1) && ((f[2,1] % f[1,1]) == 1), print1(n, ", ")););} \\ Michel Marcus, Sep 28 2017

list(lim)=my(v=List()); if(lim<9, return([])); forprime(p=3,sqrtint(((lim\=1)-1)\2), forprimestep(q=2*p+1,lim,2*p, listput(v, p*q))); Set(v) \\ Charles R Greathouse IV, Feb 08 2021

More terms from Jinyuan Wang, May 16 2020

A086792 Orders of finite groups G with the property that the sum of the orders of all the proper normal subgroups of G equals the order of G.

Original entry on oeis.org

6, 12, 28, 30, 56, 360, 364, 380, 496, 760, 792, 900, 992, 1224, 1656, 1680, 1980

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 04 2003

The only Abelian groups with this property are the cyclic groups C_n where n is a perfect number, so this sequence can be seen as a groups analogy of perfect numbers.

Derek Holt (mareg(AT)mimosa.csv.warwick.ac.uk) computed the orders of the non-Abelian groups in the sequence up to n=500 and commented "In general, if 2^n - 1 is a Mersenne prime, then 2^(n-1)*(2^n - 1) is a perfect number and the group with presentation < x,y | x^(2^n-1) = 1, y^(2^n) = 1, y^-1 x y = x^-1 > has order equal to the sum of the orders of its proper normal subgroups." So if n is an even perfect number, 2n also belongs to this sequence (the numbers 12 and 56 above).

Tom Leinster, Perfect numbers and groups, arXiv:math/0104012 [math.GR], 2001.
Tom De Medts and Attila Maróti, Perfect numbers and finite groups
Tom De Medts, Leinster groups. [archived]
Sci.math, Groups analogy of perfect numbers

Subsequence of A060652.

Cf. A000396.

a(10)-a(17) added using "Leinster groups" link by Eric M. Schmidt, May 02 2014

Showing 1-8 of 8 results.

cp's OEIS Frontend

A386233 Number of good involutions of all nontrivial conjugation quandles of order A060652(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

A051532 The abelian orders (or abelian numbers): numbers m such that every group of order m is abelian.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

A056866 Orders of non-solvable groups, i.e., numbers that are not solvable numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A064899 Numbers that are of the form pq where p and q are distinct primes that are the orders of non-Abelian groups.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Extensions

A056868 Numbers that are not nilpotent numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A068919 Squarefree orders of non-Abelian groups.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

A221048 The odd semiprime numbers (A046315) which are orders of a non-Abelian group.

Views

Author

Keywords

Comments

Links