A000019 -id:A000019 - cp's OEIS frontend

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

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

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

A109471 Cumulative sum of absolute values of coefficients of q^(2n) in the series expansion of Ramanujan's mock theta function f(q).

Original entry on oeis.org

1, 3, 6, 11, 17, 27, 38, 55, 76, 103, 136, 182, 235, 303, 385, 489, 612, 766, 945, 1166, 1428, 1742, 2111, 2557, 3072, 3686, 4401, 5246, 6223, 7371, 8692, 10236, 12014, 14074, 16435, 19171, 22292, 25884, 29981, 34677, 40017, 46122, 53038, 60920

Offset: 0

Jonathan Vos Post, Aug 28 2005

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000039, A000019, A053250, A053251, A053252, A053253, A053254, A053255, A064053.

nmax = 200; f[q_, s_] := Sum[q^(n^2)/Product[1 + q^k, {k, n}]^2, {n, 0, s}]; A000039:= CoefficientList[Series[f[q, nmax], {q, 0, nmax}], q][[1 ;; -1 ;; 2]]; Table[Sum[Abs[A000039[[k]]], {k,1,n}], {n,1,51}] (* G. C. Greubel, Feb 18 2018 *)

a(n) = Sum_{k=0..n} abs(A000039(k)). [corrected by Joerg Arndt, Feb 25 2018]

a(n) ~ sqrt(3/2) * exp(sqrt(n/3)*Pi) / Pi. - Vaclav Kotesovec, Jun 12 2019

A104109 Numbers n such that whenever a group G has a solvable subgroup of index n, then G itself is solvable.

Original entry on oeis.org

1, 2, 3, 4, 11, 19, 22, 23, 29, 31, 37, 41, 43, 46, 47, 53, 57, 58, 59, 61, 67, 69, 71, 73, 79, 83, 86, 87, 89, 92, 93, 94, 97, 101, 103, 106, 107, 109, 111, 113, 116, 118

Offset: 1

David L. Harden, Mar 04 2005

The way the sequence is defined, it is not obvious that it is computable.
The following condition is obviously computable and equivalent to the criterion for membership in this sequence: n is in the sequence iff it is true that a transitive subgroup of the symmetric group S_n is solvable iff it has a solvable point stabilizer.
As for the primes in the sequence, a theorem proved by Burnside guarantees that any unsolvable permutation group on a prime number of points is doubly transitive. The classification of doubly transitive permutation groups has been completed and it can be used to show that the only primes not in this sequence are 7, 13 and the Fermat primes larger than 3. So assuming the standard conjecture that 65537 is the largest Fermat prime, all primes except for these six (5, 7, 13, 17, 257 and 65537) are in this sequence.
If q (not equal to 2 or 3) is a power of a prime, then q+1 is not in this sequence, since PGL(2,q) is an unsolvable group with a solvable subgroup of index q+1 (isomorphic to the general affine group GA(1,q)).
Finally, if n is a nonmember of this sequence, then so is any multiple of n:
Since n is a nonmember of this sequence, there is a group G such that G is unsolvable and G has a solvable subgroup H of index n. Suppose N=d*n is a multiple of n. Then G x C_d (here C_d denotes a cyclic group of order d) is a group which is unsolvable because G is unsolvable. Nevertheless, it has a subgroup of index N isomorphic to H, which is solvable: this is the H x 1 subgroup.
This also means that any divisor of a member of this sequence is in this sequence and this form of the above-proved property is useful in proving membership in the sequence. One nice application of this form of that property is the following:
If n is insipid (that is, n is in A102842) and n/p is in this sequence for all primes p dividing n, then n is in this sequence.
Proof. Since n is insipid, n=1,2,3 or 4 or n>=34. If n=1,2,3 or 4, this is trivial since all of those values of n are in this sequence.
If n>=34, then let G be a group and let H be a solvable subgroup of index n in G.
Claim. H is not maximal in G.
Proof of Claim. Suppose H is maximal in G. Then let N be the core of H in G. G/N is isomorphic to a primitive permutation group on n points. Since n is insipid, this means G/N ~= A_n or S_n. Then restricting the quotient map (the map from G to G/N) to H yields a surjective homomorphism from H to a point stabilizer in S_n or A_n. This point stabilizer is isomorphic to S_(n-1) or A_(n-1). Since n>=34, n-1 >= 33 >= 5 so this image of H under this homomorphism is unsolvable and therefore H is unsolvable for a contradiction. The Claim is proved.
Since H is not maximal in G, there is a subgroup K of G such that G is properly contained in G and properly contains H. Then, by Lagrange's Theorem, [G:K] and [K:H] are proper divisors of n. This means there are (not necessarily distinct) prime factors p,q of n such that [G:K] | n/p and [K:H] | n/q. By our assumption, n/p and n/q are in this sequence. Then the property proved earlier implies that [G:K] and [K:H] are in this sequence. This means that the solvability of H implies the solvability of K, which, in turn, implies the solvability of G and the membership of n in this sequence. The proof is now complete.

			22 is in this sequence, since the only primitive permutation groups on 22 points are M_22, Aut(M_22), A_22 and S_22. All of these have unsolvable point stabilizers.
This shows that a counterexample to 22 being in the sequence will be (by basic properties of the action-on-cosets homomorphism) an unsolvable group G with a solvable _nonmaximal_ subgroup H of index 22. Since H is nonmaximal, there is a subgroup K lying properly between G and H. Then by Lagrange's Theorem, we know [G:K]=2 and [K:H]=11 or vice versa. In either case, since 2 and 11 are both in this sequence, the solvability of H implies the solvability of K which implies the solvability of G. Thus 22 is in the sequence.

Cf. A000019, A102842.

A132221 Number of imprimitive transitive permutation groups of degree n.

Original entry on oeis.org

0, 0, 0, 3, 0, 12, 0, 43, 23, 36, 0, 295, 0, 59, 98, 1932, 0, 979, 0, 1113, 155, 55, 0, 24995, 183, 89, 2377, 1840, 0, 5708, 0, 2801317, 158, 113, 401, 121257, 0, 72, 304, 315834, 0, 9487, 0, 2109, 10914, 54, 0

Offset: 1

Artur Jasinski, Aug 14 2007

Smallest degree of an imprimitive group is 4. The groups of degree 4 are C_4, V_4, D_4.

Imprimitive groups do not exist with prime degrees.

Cf. A000019, A002106.

a(n) = A002106(n) - A000019(n).

More terms from Vaclav Kotesovec, Jul 18 2022

A173407 Partial sums of A002106.

Original entry on oeis.org

1, 2, 4, 9, 14, 30, 37, 87, 121, 166, 174, 475, 484, 547, 651, 2605, 2615, 3598, 3606, 4723, 4887, 4946, 4953, 29953, 30164, 30260, 32652, 34506, 34514, 40226, 40238, 2841562, 2841724, 2841839, 2842246, 2963525, 2963536, 2963612, 2963918, 3279760, 3279770, 3289261, 3289271, 3291384, 3302307, 3302363, 3302369

Offset: 1

Jonathan Vos Post, Feb 17 2010

Partial sums of number of transitive permutation groups of degree n. The subsequence of primes in this partial sum begins: 2, 37, 547, 4723. The subsequence of squares in this partial sum begins: 1, 4, 9, 121, 484.

Cf. A000001, A000019, A002106, A173397.

Cases[Import["https://oeis.org/A002106/b002106.txt", "Table"], {, }][[All, 2]] // Accumulate (* Jean-François Alcover, Jan 03 2020 *)

a(n) = Sum_{i=1..n} A002106(i).

More terms from Jinyuan Wang, Feb 23 2020

More terms from Vaclav Kotesovec, Jul 18 2022

A243407 Decimal expansion of Pálfy's constant c_3 = 5/3 + log_9(32).

Original entry on oeis.org

3, 2, 4, 3, 9, 9, 1, 0, 5, 0, 5, 9, 5, 3, 1, 0, 2, 5, 9, 4, 1, 5, 4, 8, 4, 4, 5, 2, 5, 2, 3, 5, 6, 8, 8, 0, 2, 4, 1, 5, 6, 3, 0, 7, 6, 6, 9, 9, 6, 3, 6, 7, 7, 3, 6, 3, 4, 3, 3, 0, 4, 0, 2, 6, 2, 6, 3, 3, 7, 9, 6, 7, 0, 1, 1, 8, 9, 5, 3, 6, 7, 9, 3, 1, 9

Offset: 1

Charles R Greathouse IV, Jun 04 2014

Pálfy proved there are no primitive solvable permutation groups T with order greater than n^c_3 / 24^(1/3) but infinitely many for which equality is attained, where n is the degree of the group. Such groups necessarily have degree which is a power of 3, hence the subscript. He also gave tighter bounds for other prime powers.

			E(9) : 2S_4 is a primitive solvable permutation group of degree 9 and order 432 = 9^(5/3 + log_9(32))/24^(1/3).

P. P. Pálfy, A polynomial bound for the orders of primitive solvable groups, J. Algebra 77:1 (1982), pp. 127-137.
Index entries for transcendental numbers

Cf. A000019.

RealDigits[5/3+Log[9,32],10,120][[1]] (* Harvey P. Dale, Mar 05 2015 *)

PARI
```
5/3+log(32)/log(9)
```

A250219 Number of times that n appears in n-th OEIS sequence (A_n), or -1 if n appears infinitely many times.

Original entry on oeis.org

-1, -1, 8, 0, -1, 4, 0, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Eric Chen, Dec 24 2014

Does a(19) equal -1? (See A000019.)

			From _Jianing Song_, Sep 05 2018: (Start)
a(3) = 8 since A000003(k) = 3 for k = 11, 19, 23, 27, 31, 43, 67 and 163.
a(6) = 4 since A000006(k) = 6 for k = 12, 13, 14 and 15.
a(10) = 2 since A000010(k) = 10 for k = 11 and 22. (End)

Index entries for sequences whose definition involves A_n (or An).

Cf. A000001, A000002, A000003, A000004, etc.

a(3) corrected by Jianing Song, Sep 05 2018

cp's OEIS Frontend

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

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

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A109471 Cumulative sum of absolute values of coefficients of q^(2n) in the series expansion of Ramanujan's mock theta function f(q).

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A104109 Numbers n such that whenever a group G has a solvable subgroup of index n, then G itself is solvable.

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A132221 Number of imprimitive transitive permutation groups of degree n.

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A173407 Partial sums of A002106.

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A243407 Decimal expansion of Pálfy's constant c_3 = 5/3 + log_9(32).

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A250219 Number of times that n appears in n-th OEIS sequence (A_n), or -1 if n appears infinitely many times.

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