A056867 -id:A056867 - cp's OEIS frontend

A003277 Cyclic numbers: k such that k and phi(k) are relatively prime; also k such that there is just one group of order k, i.e., A000001(k) = 1.

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, 145, 149, 151, 157, 159, 161, 163, 167, 173

Offset: 1

N. J. A. Sloane and Richard Stanley

Except for a(2)=2, all the terms in the sequence are odd. This is because of the existence of a non-cyclic dihedral group of order 2n for each n>1. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 09 2001
Also gcd(n, A051953(n)) = 1. - Labos Elemer
n such that x^n == 1 (mod n) has no solution 2 <= x <= n. - Benoit Cloitre, May 10 2002
There is only one group (the cyclic group of order n) whose order is n. - Gerard P. Michon, Jan 08 2008  [This is a 1947 result of Tibor Szele. - Charles R Greathouse IV, Nov 23 2011]
Any divisor of a Carmichael number (A002997) must be odd and cyclic. Conversely, G. P. Michon conjectured (c. 1980) that any odd cyclic number has at least one Carmichael multiple (if the conjecture is true, each of them has infinitely many such multiples). In 2007, Michon & Crump produced explicit Carmichael multiples of all odd cyclic numbers below 10000 (see link, cf. A253595). - Gerard P. Michon, Jan 08 2008
Numbers n such that phi(n)^phi(n) == 1 (mod n). - Michel Lagneau, Nov 18 2012
Contains A000040, and all members of A006094 except 6. - Robert Israel, Jul 08 2015
Number m such that n^n == r (mod m) is solvable for any r. - David W. Wilson, Oct 01 2015
Numbers m such that A074792(m) = m + 1. - Thomas Ordowski, Jul 16 2017
Squarefree terms of A056867 (see McCarthy link p. 592 and similar comment with "cubefree" in A051532). - Bernard Schott, Mar 24 2022

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
J. S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 7.
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..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Max Alekseyev, Michon's conjecture (Open Problem Garden, Aug. 2007).
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025. See pp. 1, 2, 30.
Keith Conrad, When are all groups of order n cyclic?, University of Connecticut, 2019.
Peter J. Dukes and Joanna Niezen, Pairwise balanced designs of dimension three, Australasian Journal Of Combinatorics, Volume 61(1) (2015), pages 98-113.
Paul Erdős, Some asymptotic formulas in number theory, J. Indian Math. Soc. (N.S.) 12 (1948), pp. 75-78.
Jose M. Grau, Manuel Rodríguez, A. Oller-Marcen, and Daniel Sadornil, Fermat test with gaussian base and Gaussian pseudoprimes, arXiv preprint arXiv:1401.4708 [math.NT], 2014.
Donald J. McCarthy, A survey of partial converses to Lagrange's theorem on finite groups, Transactions of the New York Academy of Sciences, Vol. 33, No. 6, Series II (1971), pp. 586-594. See page 592.
Gérard P. Michon, Carmichael Divisors
Gérard P. Michon and J. K. Crump, Carmichael Multiples of Odd Cyclic Numbers (up to 10000)
Jonathan Pakianathan and Krishnan Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, August-September 2000, 631-634.
R. P. Stanley, Letter to N. J. A. Sloane, c. 1991
T. Szele, Über die endlichen Ordnungszahlen, zu denen nur eine Gruppe gehört, Commentarii Mathematici Helvetici 20 (1947), pp. 265-267.
Index entries for sequences related to groups

Subsequence of A051532. Intersection of A056867 and A005117.
Cf. A000010, A008966, A009195, A050384 (the same sequence but with the primes removed). Also A000001(a(n)) = 1.
Cf. A002997, A006094, A054395, A055561, A054396, A054397, A056867, A135850, A249550, A249551, A249552, A249553, A249554, A249555, A036537, A051953, A253595.

import Data.List (elemIndices)
a003277 n = a003277_list !! (n-1)
a003277_list = map (+ 1) $ elemIndices 1 a009195_list
-- Reinhard Zumkeller, Feb 27 2012

[n: n in [1..200] | Gcd(n, EulerPhi(n)) eq 1]; // Vincenzo Librandi, Jul 09 2015

select(t -> igcd(t, numtheory:-phi(t))=1, [$1..1000]); # Robert Israel, Jul 08 2015

Select[Range[175], GCD[#, EulerPhi[#]] == 1 &] (* Jean-François Alcover, Apr 04 2011 *)
Select[Range@175, FiniteGroupCount@# == 1 &] (* Robert G. Wilson v, Feb 16 2017 *)
Select[Range[200],CoprimeQ[#,EulerPhi[#]]&] (* Harvey P. Dale, Apr 10 2022 *)

isA003277(n) = gcd(n,eulerphi(n))==1 \\ Michael B. Porter, Feb 21 2010

# Compare A050384.
def isPrimeTo(n, m): return gcd(n, m) == 1
def isCyclic(n): return isPrimeTo(n, euler_phi(n))
[n for n in (1..173) if isCyclic(n)] # Peter Luschny, Nov 14 2018

n = p_1*p_2*...*p_k (for some k >= 0), where the p_i are distinct primes and no p_j-1 is divisible by any p_i.
A000001(a(n)) = 1.
Erdős proved that a(n) ~ e^gamma n log log log n, where e^gamma is A073004. - Charles R Greathouse IV, Nov 23 2011
A000005(a(n)) = 2^k. - Carlos Eduardo Olivieri, Jul 07 2015
A008966(a(n)) = 1. - Bernard Schott, Mar 24 2022

More terms from Christian G. Bower

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

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

A059773 Maximum size of Aut(G) where G is a finite group of order n.

Original entry on oeis.org

1, 1, 2, 6, 4, 6, 6, 168, 48, 20, 10, 24, 12, 42, 8, 20160, 16, 432, 18, 40, 42, 110, 22, 336, 480, 156, 11232, 84, 28, 120, 30, 9999360, 20, 272, 24, 864, 36, 342, 156, 672, 40, 252, 42, 220, 192, 506, 46, 40320, 2016, 12000, 32, 312, 52, 303264, 110, 1008

Offset: 1

Victor S. Miller, Feb 21 2001

If n = 2^k then take G to be (Z/2Z)^k, the Abelian group with n=2^k elements and characteristic two. It is generated by any k linearly independent (non-identity) elements, so the automorphism group has size (n-1)(n-2)(n-4)...(n-2^(k-1)), which grows as n^log n. I think one can show that this is optimal for n=2^k and furthermore that this has the highest rate of growth for any infinite sequence of n's. - Michael Kleber, Feb 21 2001

Equals A061350(n) for n in A056867. - Eric M. Schmidt, Mar 02 2013

			The corresponding groups are 1, Z2, Z3, (Z2)^2, Z5, S3, Z7, (Z2)^3, (Z3)^2, D5, Z11, A4, Z13, D7, Z15, (Z2)^4, Z17, ...

Eric M. Schmidt, Table of n, a(n) for n = 1..767

Cf. A061350.

A059773 := function(n) local max, f, i; if IsPrimePowerInt(n) then f := PrimePowersInt(n); return Product([0..f[2]-1], k->n-f[1]^k); fi; max := 1; for i in [1..NumberSmallGroups(n)] do max := Maximum(max, Size(AutomorphismGroup(SmallGroup(n,i)))); od; return max; end; # Eric M. Schmidt, Mar 02 2013

More terms from Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 09 2001

a(18)-a(56) from Stephen A. Silver, Feb 26 2013

A066060 Number of nilpotent groups of order n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 5, 2, 1, 1, 2, 1, 1, 1, 14, 1, 2, 1, 2, 1, 1, 1, 5, 2, 1, 5, 2, 1, 1, 1, 51, 1, 1, 1, 4, 1, 1, 1, 5, 1, 1, 1, 2, 2, 1, 1, 14, 2, 2, 1, 2, 1, 5, 1, 5, 1, 1, 1, 2, 1, 1, 2, 267, 1, 1, 1, 2, 1, 1, 1, 10, 1, 1, 2, 2, 1, 1, 1, 14, 15, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 51, 1, 2, 2, 4, 1

Offset: 1

Reiner Martin, Dec 29 2001

Multiplicative with a(p^m) equal to the number of groups of order p^m.

T. D. Noe and Amiram Eldar, Table of n, a(n) for n = 1..2047 (terms 1..2015 from T. D. Noe)
John Renze, Nilpotent Group.

Cf. A000001, A056867, A069739.

terms = 101; fgc = FiniteGroupCount[Range[terms]]; a[1] = 1; a[n_ /; PrimePowerQ[n] && 1 < n <= terms] := a[n] = fgc[[n]]; a[n_ /; 1 < n <= terms] := a[n] = Times @@ (a[#[[1]]^#[[2]]]& /@ FactorInteger[n]); Array[a, terms] (* Jean-François Alcover, Oct 03 2017 *)

A357900 Number of groups of order A060702(n) with trivial center.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 5, 1, 1, 1, 1, 3, 1, 1, 1, 1, 6, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 5, 2, 5, 1, 1, 5, 2, 1, 2, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 3, 1, 4, 1, 1, 4, 1, 1, 17, 1, 1, 5, 1, 1, 1, 1, 8, 1, 1, 2, 1, 11, 1, 2, 2, 5, 1, 1, 1, 2, 1, 1, 3, 1, 1, 19

Offset: 1

Jianing Song, Oct 19 2022

Among the data currently known, it seems that the indices of records are n's such that A060702(n) = 1, 18, 54, 72, 162, 216, 486, 648, 972, 1458, ... with record values 1, 2, 5, 6, 17, 19, 72, 79, 109, 443, ...

			a(2) = 1 since there is a unique group of order A060702(2) = 6 with trivial center: S3.

Jianing Song, Table of n, a(n) for n = 1..372
Jianing Song, Number of centerless groups of order n <= 2022 (skipping n = 768, 1152, 1280, 1536, 1728, 1792, 1920, 2016)

Cf. A060702, A059806, A056867.

IsNilpotentNumber := function(n) # if n > 1 is a nilpotent number, then no group of order n has trivial center; see also A056867
    local c, omega, i, j;
    c := PrimePowersInt( n );
    omega := Length(c)/2;
    for i in [1..omega] do
        for j in [1..c[2*i]] do
            if GcdInt(n, c[2*i-1]^j-1) > 1 then
                return false;
            fi;
        od;
    od;
    return true;
end;
CountTrivialCenter := function(n) # returns the number of groups of order n with trivial center
    local count, i;
    if n > 1 and IsNilpotentNumber(n) then
        return 0;
    fi;
    count := 0;
    for i in [1..NumberSmallGroups(n)] do
        if(Size(Center(SmallGroup(n, i))) = 1) then
            count:=count+1;
        fi;
    od;
    return count;
end;

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))

A335419 Integers m such that every group of order m is not simple.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 21, 20, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94

Offset: 1

Bernard Schott, Jul 09 2020

nonn

Officially, the group of order 1 is not considered to be simple; "a group <> 1 is simple if it has no normal subgroups other than G and 1" (See reference for Joseph J. Rotman's definition).
There is no prime term because there exists only one group of order p and this cyclic group Z/pZ is simple.
As a consequence of Feit-Thompson theorem, all odd composites are terms of this sequence.
The first composite even number that is not present in the data is 60 that is the order of simple alternating group Alt(5), the second one that is missing is 168 corresponding to simple Lie group PSL(3,2) [A031963].

			There exist 5 (nonisomorphic) groups of order 8: Z/8Z, Z/2Z × Z/4Z, (Z/2Z)^3, D_4 and H_8; none of these 5 groups is simple, so 8 is a term.
There exist 13 (nonisomorphic) groups of order 60 (see A000001), 12 are not simple but the alternating group Alt(5) is simple, hence 60 is not a term.

Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, Exercice 1.44 p.96.
Joseph J. Rotman, The Theory of Groups: An Introduction, 4th ed., Springer-Verlag, New-York, 1995. Page 39, Definition.

Craig Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574-577.
Wikipedia, Feit-Thompson theorem.

Complement of A005180 (except for 1).
Subsequence: A014076 (odd nonprimes).
Cf. A000001, A031963, A051532 (similar for Abelian), A056867 (similar for nilpotent).

cp's OEIS Frontend

A003277 Cyclic numbers: k such that k and phi(k) are relatively prime; also k such that there is just one group of order k, i.e., A000001(k) = 1.

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

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

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A056868 Numbers that are not nilpotent numbers.

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A059773 Maximum size of Aut(G) where G is a finite group of order n.

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A066060 Number of nilpotent groups of order n.

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A357900 Number of groups of order A060702(n) with trivial center.