A005180 -id:A005180 - cp's OEIS frontend

A002229 Primitive roots that go with the primes in A002230.

1, 2, 3, 5, 6, 7, 19, 21, 23, 31, 37, 38, 44, 69, 73, 94, 97, 101, 107, 111, 113, 127, 137, 151, 164, 179, 194, 197, 227, 229, 263, 281, 293, 335, 347, 359, 401, 417

Offset: 1

N. J. A. Sloane

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. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. XLIV.

R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Kevin J. McGown and Jonathan P. Sorenson, Computation of the least primitive root, arXiv:2206.14193 [math.NT], 2022.
A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots, Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968 [Annotated scans of selected pages]

Cf. A002230.

s = {1}; rm = 1; Do[p = Prime[k]; r = PrimitiveRoot[p]; If[r > rm, Print[r]; AppendTo[s, r]; rm = r], {k, 10^6}]; s (* Jean-François Alcover, Apr 05 2011 *)

from sympy import isprime, primitive_root
from itertools import count, islice
def f(n): return 0 if not isprime(n) or (r:=primitive_root(n))==None else r
def agen(r=0): yield from ((m, r:=f(m))[1] for m in count(1) if f(m) > r)
print(list(islice(agen(), 15))) # Michael S. Branicky, Feb 13 2023

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

a(35)-a(38), from McGown and Sorenson, added by Michel Marcus, Jun 29 2022

A000381 Essentially the same as A001611.

Original entry on oeis.org

2, 3, 4, 6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766

Offset: 0

N. J. A. Sloane, Apr 17 2015

dead

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).

R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Cf. A001611.

A000381:=-(-2+z+2*z**2)/(z-1)/(z**2+z-1); # [Simon Plouffe in his 1992 dissertation.]

A005348 Number of ways to add n ordinals.

Original entry on oeis.org

1, 2, 5, 13, 33, 81, 193, 449, 1089, 2673, 6561, 15633, 37249, 88209, 216513, 531441, 1266273, 3017169, 7189057, 17537553, 43046721, 102568113, 244390689, 582313617, 1420541793, 3486784401, 8308017153, 19795645809, 47167402977, 115063885233, 282429536481

Offset: 1

N. J. A. Sloane, R. K. Guy, Bill Sands and Tommy Kucera

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 270-271.
W. Sierpiński, Cardinal and Ordinal Numbers, 2nd ed. p 275.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. Erdős, Some remarks on set theory, Proc. Am. Math. Soc. 1 (1950) 127-141.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Bill Sands and Tommy Kucera, Letter to N. J. A. Sloane, Jun 10 1975.
A. Wakulicz, Sur la somme d'un nombre fini de nombres ordinaux, Fund. Math. 36 (1949), 254-266.
Eric Weisstein's World of Mathematics, Ordinal Number
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,81).

I:=[1,2,5,13,33,81,193,449,1089,2673,6561,15633,37249, 88209,216513,531441,1266273,3017169,7189057,17537553, 43046721]; [n le 21 select I[n] else 81*Self(n-5): n in [1..50]]; // Vincenzo Librandi, Dec 17 2015

Join[{1, 2, 5, 13, 33, 81, 193, 449, 1089, 2673, 6561, 15633, 37249, 88209}, LinearRecurrence[ {0, 0, 0, 0, 81}, {216513, 531441, 1266273, 3017169, 7189057}, 20]] (* Harvey P. Dale, Dec 15 2014 *)

a(n) = 81*a(n-5) for n >= 21.

A005707 a(1) = a(2) = a(3) = a(4) = 1, a(n) = a(a(n-1))+a(n-a(n-1)) for n >= 5.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 7, 8, 8, 9, 9, 9, 9, 9, 9, 10, 11, 11, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 14, 15, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 20, 21, 22, 22, 23, 24, 24, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 27, 27

Offset: 1

N. J. A. Sloane

It is known that a(n)-a(n-1)=0 or 1 (see the 1991 Monthly reference). - Emeric Deutsch, Jun 06 2005

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
D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.
R. K. Guy, Letter to N. J. A. Sloane with attachment, 1988
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
D. Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.

Cf. A004001, A005350.

Cf. A005185.

a005707 n = a005707_list !! (n-1)
a005707_list = 1 : 1 : 1 : 1 : h 5 1 where
   h x y = z : h (x + 1) z where z = a005707 y + a005707 (x - y)
-- Reinhard Zumkeller, Jul 20 2012

a[1]:=1: a[2]:=1: a[3]:=1: a[4]:=1: for n from 5 to 100 do a[n]:=a[a[n-1]]+a[n-a[n-1]] od: seq(a[n],n=1..100); # Emeric Deutsch, Jun 06 2005

a[1]=a[2]=a[3]=a[4]=1;a[n_]:=a[n]=a[a[n-1]]+a[n-a[n-1]];Table[a[i],{i,80}] (* Harvey P. Dale, Jan 22 2013 *)

More terms from Emeric Deutsch, Jun 06 2005

A001051 Number of subgroups of order n in orthogonal group O(3).

Original entry on oeis.org

1, 3, 1, 5, 1, 5, 1, 7, 1, 5, 1, 8, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 10, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 8, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 8, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 8

Offset: 1

J. H. Conway

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Orthogonal Group.

The main sequences concerned with group theory are A000001, A000679, A001034, A001051, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A051881.

a[2] = 3; a[4] = 5; a[12] = 8; a[24] = 10; a[48] = a[60] = a[120] = 8; a[n_] := Switch[Mod[n, 4], 0, 7, 1, 1, 2, 5, 3, 1]; Table[a[n], {n, 1, 96}] (* Jean-François Alcover, Oct 15 2013 *)

A001051(n) = if((12==n)||(48==n)||(60==n)||(120==n),8,if(24==n,10,if((4==n)||(2==n),1+n,[1,5,1,7][1+((n-1)%4)]))); \\ Antti Karttunen, Jan 15 2019

Has period 1 5 1 7 except that a(2) = 3, a(4) = 5, a(12) = 8, a(24) = 10, a(48) = a(60) = a(120) = 8.

Data section extended up to a(120) by Antti Karttunen, Jan 15 2019

A005182 a(n) = floor(e^((n-1)/2)).

Original entry on oeis.org

0, 1, 1, 2, 4, 7, 12, 20, 33, 54, 90, 148, 244, 403, 665, 1096, 1808, 2980, 4914, 8103, 13359, 22026, 36315, 59874, 98715, 162754, 268337, 442413, 729416, 1202604, 1982759, 3269017, 5389698, 8886110, 14650719, 24154952, 39824784

Offset: 0

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

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy and N. J. A. Sloane, Correspondence, 1988.

Floor[E^((Range[0,40]-1)/2)] (* Harvey P. Dale, Mar 20 2016 *)

A051881 Number of subgroups of order n in special orthogonal group SO(3).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1

Offset: 1

J. H. Conway

			The groups are "nn", of order n; "22n", of order 2n; "332", "432", "532" of orders 12,24,60.

Antti Karttunen, Table of n, a(n) for n = 1..12000

The main sequences concerned with group theory are A000001, A000679, A001034, A001051, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A051881.

a[2] = 1; a[12|24|60] = 3; a[n_] := 2-Mod[n, 2]; Array[a, 105] (* Jean-François Alcover, Nov 12 2015 *)

a(n)=if(n==2||n==12||n==24||n==60, if(n>2,3,1), if(n%2,1,2)) \\ Charles R Greathouse IV, Nov 10 2015

def a(n):
    if n == 2:
        return 1
    elif n in {12, 24, 60}:
        return 3
    else:
        return 2 - n % 2 # Paul Muljadi, Oct 21 2024

Has period 1, 2 except for a(2) = 1, a(12) = a(24) = a(60) = 3.

More terms from James Sellers and David W. Wilson, Dec 16 1999

A119648 Orders for which there is more than one simple group.

Original entry on oeis.org

20160, 4585351680, 228501000000000, 65784756654489600, 273457218604953600, 54025731402499584000, 3669292720793456064000, 122796979335906113871360, 6973279267500000000000000, 34426017123500213280276480

Offset: 1

N. J. A. Sloane, Jul 29 2006

nonn

All such orders are composite numbers (since there is only one group of any prime order).
Orders which are repeated in A109379.
Except for the first number, these are the orders of symplectic groups C_n(q)=Sp_{2n}(q), where n>2 and q is a power of an odd prime number (q=3,5,7,9,11,...). Also these are the orders of orthogonal groups B_n(q). - Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010
a(1) = 20160 = 8!/2 is the order of the alternating simple group A_8 that is isomorphic to the Lie group PSL_4(2), but 20160 is also the order of the Lie group PSL_3(4) that is not isomorphic to A_8 (see A137863). - Bernard Schott, May 18 2020

			From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010: (Start)
a(1)=|A_8|=8!/2=20160,
a(2)=|C_3(3)|=4585351680,
a(3)=|C_3(5)|=228501000000000, and
a(4)=|C_4(3)|=65784756654489600. (End)

See A001034 for references and other links.
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]. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]

C. Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574-577. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory. See also. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
W. Kimmerle et al., Composition Factors from the Group Ring and Artin's Theorem on Orders of Simple Groups, Proc. London Math. Soc., (3) 60 (1990), 89-122. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
David A. Madore, Orders of non-Abelian simple groups
Wikipedia, Classification of finite simple groups [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
Index entries for sequences related to groups

Cf. A000001, A000679, A005180, A001228, A060793, A056866, A056868, A119630.

Cf. A001034 (orders of simple groups without repetition), A109379 (orders with repetition), A137863 (orders of simple groups which are non-cyclic and non-alternating).

sp(n, q) 1/2 q^n^2.(q^(2.i) - 1, i, 1, n) [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010] [This line contained some nonascii characters which were unreadable]

For n>1, a(n) is obtained as (1/2) q^(m^2)Prod(q^(2i)-1, i=1..m) for appropriate m>2 and q equal to a power of some odd prime number. [Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]

Extended up to the 10th term by Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010

A137863 Orders of simple groups which are non-cyclic and non-alternating.

Original entry on oeis.org

168, 504, 660, 1092, 2448, 3420, 4080, 5616, 6048, 6072, 7800, 7920, 9828, 12180, 14880, 20160, 25308, 25920, 29120, 32736, 34440, 39732, 51888, 58800, 62400, 74412, 95040, 102660, 113460, 126000, 150348, 175560, 178920, 194472, 246480, 262080

Offset: 1

Artur Jasinski, Feb 16 2008

nonn

From Bernard Schott, Apr 26 2020: (Start)
About a(16) = 20160; 20160 = 8!/2 is the order of the alternating simple group A_8 that is isomorphic to the Lie group PSL_4(2), but, 20160 is also the order of the Lie group PSL_3(4) that is not isomorphic to A_8.
Indeed, 20160 is the smallest order for which there exist two nonisomorphic simple groups and it is the order of this group PSL_3(4) that was missing in the data. The first proof that there exist two nonisomorphic simple groups of this order was given by the American mathematician Ida May Schottenfels (1900) [see the link]. (End)

			From _Bernard Schott_, Apr 27 2020: (Start)
Two particular examples:
a(1) = 168 is the order of the smallest non-cyclic and non-alternating simple group, this Lie group is the projective special linear group PSL_2(7) that is isomorphic to the general linear group GL_3(2).
a(12) = 7920 is the order of the smallest sporadic group (A001228), the Mathieu group M_11. (End)

L. E. Dickson, Linear groups, with an exposition of the Galois field theory (Teubner, 1901), p. 309.

David Madore, Orders of non abelian simple groups
Ida May Schottenfels, Two non isomorphic simple groups of the same order 20160, Annals of Mathematics, Second Series, Vol. 1, No. 1/4 (1900), pp. 147-152.

Cf. A001034, A001710, A005180, A109379.

Subsequence: A001228 (sporadic groups).

More terms from R. J. Mathar, Apr 23 2009
a(16) = 20160 inserted by Bernard Schott, Apr 26 2020
Incorrect formula and programs removed by R. J. Mathar, Apr 27 2020
Terms checked by Bernard Schott, Apr 26 2020

A385030 Orders of characteristically simple groups.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 168, 169, 173, 179, 181, 191, 193, 197, 199, 211

Offset: 1

Miles Englezou, Jun 15 2025

nonn

Equivalently, orders k of groups G where a G exists as a direct product of isomorphic simple groups.

A group G is characteristically simple if it contains no characteristic proper subgroups (a subgroup which is invariant under every automorphism of G). Since a finite group is characteristically simple if and only if it is a direct product of isomorphic simple groups, G is characteristically simple if and only if it is an elementary abelian group or a direct product of isomorphic nonabelian simple groups.

			5 is a term since C_5 is prime cyclic and contains no proper subgroups. Therefore it contains no characteristic proper subgroups.
60 is a term since the alternating group A_5 is simple and contains no normal subgroups. Therefore it contains no characteristic proper subgroups.
3600 is a term since the direct product A_5 x A_5, though it contains A_5 twice as a normal subgroup and is therefore not simple, it contains no characteristic proper subgroups.

Miles Englezou, Table of n, a(n) for n = 1..10000
Wikipedia, Characteristically simple group

Cf. A001034, A005180, A246655.

isok := function(G)
    if Order(G) = 1 then
        return false;
    elif IsElementaryAbelian(G) then
        return true;
    elif IsSimpleGroup(G) then
        return true;
    else
        for K in AllSubgroups(G) do
            if IsCharacteristicSubgroup(G, K) then
                return false;
            fi;
        od;
        return true;
    fi;
end;

Union of A246655 and the nonzero powers of every term in A001034.

cp's OEIS Frontend

A002229 Primitive roots that go with the primes in A002230.

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A000381 Essentially the same as A001611.

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Maple

A005348 Number of ways to add n ordinals.

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Magma

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A005707 a(1) = a(2) = a(3) = a(4) = 1, a(n) = a(a(n-1))+a(n-a(n-1)) for n >= 5.

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A001051 Number of subgroups of order n in orthogonal group O(3).

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A005182 a(n) = floor(e^((n-1)/2)).

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Mathematica

A051881 Number of subgroups of order n in special orthogonal group SO(3).

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A119648 Orders for which there is more than one simple group.

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