A051593 -id:A051593 - cp's OEIS frontend

A057740 Irregular triangle read by rows: T(n,k) is the number of elements of alternating group A_n having order k, for n >= 1, 1 <= k <= A051593(n).

1, 1, 1, 0, 2, 1, 3, 8, 1, 15, 20, 0, 24, 1, 45, 80, 90, 144, 1, 105, 350, 630, 504, 210, 720, 1, 315, 1232, 3780, 1344, 5040, 5760, 0, 0, 0, 0, 0, 0, 0, 2688, 1, 1323, 5768, 18900, 3024, 37800, 25920, 0, 40320, 9072, 0, 15120, 0, 0, 24192

Offset: 1

Roger Cuculière, Oct 29 2000

			Triangle begins:
  1;
  1;
  1,    0,    2;
  1,    3,    8;
  1,   15,   20,     0,   24;
  1,   45,   80,    90,  144;
  1,  105,  350,   630,  504,   210,   720;
  1,  315, 1232,  3780, 1344,  5040,  5760, 0,     0,    0, 0,     0, 0, 0,  2688;
  1, 1323, 5768, 18900, 3024, 37800, 25920, 0, 40320, 9072, 0, 15120, 0, 0, 24192;
...

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].

Koda, Tatsuhiko; Sato, Masaki; Takegahara, Yugen; 2-adic properties for the numbers of involutions in the alternating groups, J. Algebra Appl. 14 (2015), no. 4, 1550052 (21 pages).
Index entries for sequences related to groups

Cf. A057731 (S_n), A054522, A057741, A051593, A000793.
Column 2, 3, 4 are A048099, A001471, A051695.
See also A061129, A061130, A061131, A061132, A061133, A061134, A061135, A061136, A061137, A061138, A061139, A061140.

Magma
```
{* Order(g) : g in Alt(6) *};
```

row[n_] := (orders = PermutationOrder /@ GroupElements[AlternatingGroup[n] ]; Table[Count[orders, k], {k, 1, Max[orders]}]); Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Aug 31 2016 *)

More terms from N. J. A. Sloane, Nov 01 2000

Missing zero in the row for A_9 inserted by N. J. A. Sloane, Mar 27 2015

A303728 Triangle read by rows: T(n,k) is the number of labeled cyclic subgroups of order k in the alternating group A_n, 1 <= k <= A051593(n).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 3, 4, 1, 15, 10, 0, 6, 1, 45, 40, 45, 36, 1, 105, 175, 315, 126, 105, 120, 1, 315, 616, 1890, 336, 2520, 960, 0, 0, 0, 0, 0, 0, 0, 336, 1, 1323, 2884, 9450, 756, 18900, 4320, 0, 6720, 2268, 0, 3780, 0, 0, 3024, 1, 5355, 15520, 47250, 19656

Offset: 1

Andrew Howroyd, Jul 03 2018

			Triangle begins:
1;
1;
1, 0, 1;
1, 3, 4;
1, 15, 10, 0, 6;
1, 45, 40, 45, 36;
1, 105, 175, 315, 126, 105, 120;
1, 315, 616, 1890, 336, 2520, 960, 0, 0, 0, 0, 0, 0, 0, 336;
...

Row sums are A051636.

Cf. A051593, A074881, A181950.

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
G(n)={my(s=0); forpart(p=n, if(sum(i=1,#p,p[i]-1)%2==0, my(d=lcm(Vec(p))); s+=x^d*permcount(p)/eulerphi(d))); s}
for(n=1, 10, print(Vecrev(G(n)/x)))

A053039 Exponent of largest power of 2 which appears in the cototient-iteration started with n!.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 9, 12, 14, 11, 13, 16, 14, 16, 23, 27, 23, 23, 24, 21, 23, 28, 42, 46, 26, 26, 26, 36, 43, 29, 50, 55, 37, 37, 40, 40, 39, 59, 39, 44, 68, 42, 42, 44, 51, 45, 50, 53, 49, 52, 51, 85, 55, 57, 53, 57, 60, 85, 62, 71, 62, 63, 60, 66, 66, 107, 67, 101, 76, 70, 75, 77

Offset: 1

Labos Elemer, Feb 24 2000

nonn

If the exponent is a(n), then the number of powers of 2 in the iteration-chain is 1+a(n), the maximal 2-power is 2^a(n) and the number of iterations (until fixed state) performed on these 2-powers is a(n).

			For n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and starting the iteration of A051593 with n!, the first powers of 2 which appear are 1, 2, 4, 16, 32, 128, 512, 4096, 16384, 2048 and the corresponding exponents are a(n) = 0, 1, 2, 4, 5, 7, 9, 12, 14, 11.

Cf. A051953, A053038, A053475.

Log2 /@ Table[NestWhile[# - EulerPhi@ # &, n!, ! IntegerQ@ Log2@ # &], {n, 60}] (* Michael De Vlieger, Aug 15 2017 *)

cototient(x)= x - eulerphi(x)
FunctionIterate(f,x,t)= {local(retval); retval = vector(0); while(x!=t, x = eval(concat(f,"(x)")); retval = concat(retval,x)); retval;}
A053039(x) = {local(li,fa,retval); count = 0; li = concat([x! ], FunctionIterate("cototient", x!, 0)); for(i=1,#li, fa = factor(li[i]); if(((matsize(fa)[1] == 1) && (fa[1,1] == 2)),retval = fa[1,2]; break)); retval}
for(i=1,72,print1(A053039(i),", ")) \\ Olaf Voß, Feb 21 2008

More terms from Olaf Voß, Feb 21 2008

A181950 Weighted sum of all cyclic subgroups of the Alternating Group A_n.

Original entry on oeis.org

1, 1, 4, 19, 91, 571, 4096, 38599, 370399, 3771751, 40020916, 486887611, 6457566259, 97397627419, 1566407932636, 25622476773391, 416792928270751, 7346982309720079, 141863542111338124, 2968348473040595971, 65223378275792128771, 1460499016109864574691, 32600807940657384926884

Offset: 1

Olivier Gérard, Apr 03 2012

nonn

Sum of the order of all cyclic subgroups of Alt_n.
Each permutation is counted as many times as it appears in a cyclic subgroup.
a(7) = 2^12 is remarkable as a power of 2.

			a(5) = 1*1 + 2*15 + 3*10 + 5*6 = 1 + 30 +30 +30 = 91.

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A051593, A051636, A303728.

\\ permcount is number of permutations of given type.
permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
a(n)={my(s=0); forpart(p=n, if(sum(i=1,#p,p[i]-1)%2==0, my(d=lcm(Vec(p))); s+=d*permcount(p)/eulerphi(d))); s} \\ Andrew Howroyd, Jul 03 2018

a(n) = Sum_{k=1..A051593(n)} k*A303728(n, k). - Andrew Howroyd, Jul 03 2018

Terms a(9) and beyond from Andrew Howroyd, Jul 03 2018

A057742 Maximal order of element of alternating group A_{2n}.

Original entry on oeis.org

1, 1, 3, 5, 15, 21, 35, 60, 105, 140, 210, 420, 420, 1155, 1365, 2310, 4620, 5460, 9240, 13860, 16380, 27720, 32760, 60060, 60060, 120120, 180180, 180180, 360360, 360360, 510510, 1021020, 1141140, 2042040, 3063060, 3423420, 6126120, 6846840

Offset: 0

N. J. A. Sloane, Oct 29 2000

Bisection of A051593. Cf. A057743.

More terms from Sean A. Irvine, Mar 25 2013

A057743 Maximal order of element of alternating group A_{2n+1}.

Original entry on oeis.org

1, 3, 5, 7, 15, 21, 35, 105, 105, 210, 420, 420, 840, 1260, 1540, 2520, 4620, 5460, 9240, 15015, 16380, 30030, 60060, 60060, 120120, 180180, 180180, 360360, 360360, 471240, 556920, 1021020, 1141140, 2042040, 3063060, 3423420, 6126120, 6846840

Offset: 0

N. J. A. Sloane, Oct 29 2000

Bisection of A051593. Cf. A057742.

More terms from Sean A. Irvine, Mar 25 2013

A355572 Largest LCM of partitions of n into odd parts.

Original entry on oeis.org

1, 1, 3, 3, 5, 5, 7, 15, 15, 21, 21, 35, 35, 45, 105, 105, 105, 105, 165, 165, 315, 315, 385, 385, 495, 1155, 1155, 1365, 1365, 1365, 1365, 3465, 3465, 4095, 4095, 5005, 5005, 6435, 15015, 15015, 15015, 15015, 19635, 19635, 45045, 45045, 45045, 45045, 58905, 58905, 69615, 69615

Offset: 1

Torsten Muetze, Jul 07 2022

nonn

The largest LCM is attained for a partition of n into powers of distinct odd primes and 1's.

			The partitions of n=8 into odd parts are 7+1, 5+3, 5+1+1+1, 3+3+1+1, 3+1+1+1+1+1, 1+1+1+1+1+1+1+1, and the partition with largest LCM among those is 5+3, which has LCM(5,3)=5*3=15, so a(8)=15.

Petr Gregor, Arturo Merino, and Torsten Mütze, The Hamilton compression of highly symmetric graphs, arXiv preprint arXiv:2205.08126 [math.CO], 2022.

Cf. A000793, A051593, A355573, A159685.

a(n) = my(x=1); forpart(p=n, if (!#select(x->((x%2)==0), Vec(p)), x = max(x, lcm(Vec(p))))); x; \\ Michel Marcus, Jul 08 2022

A355573 Largest LCM of partitions of n with a nonzero even number of even parts.

Original entry on oeis.org

2, 2, 4, 6, 6, 12, 12, 20, 30, 30, 60, 60, 84, 84, 140, 210, 210, 420, 420, 420, 420, 840, 840, 1260, 1260, 1540, 2310, 2520, 4620, 4620, 5460, 5460, 9240, 9240, 13860, 13860, 16380, 16380, 27720, 30030, 32760, 60060, 60060, 60060, 60060, 120120, 120120, 180180, 180180, 180180, 180180

Offset: 4

Torsten Muetze, Jul 07 2022

nonn

The largest LCM is attained for a partition of n into powers of distinct odd primes, 2^k for some k>0, 2, and 1's.

			The partitions of n=8 with a nonzero even number of even parts are 6+2, 4+4, 4+2+1+1, 3+2+2+1, 2+2+2+2, 2+2+1+1+1+1, and the partition with largest LCM among those is 3+2+2+1, which has LCM(3,2,2,1)=3*2=6, so a(8)=6.

Petr Gregor, Arturo Merino, and Torsten Mütze, The Hamilton compression of highly symmetric graphs, arXiv preprint arXiv:2205.08126 [math.CO], 2022.

Cf. A000793, A051593, A355572, A159685.

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A057740 Irregular triangle read by rows: T(n,k) is the number of elements of alternating group A_n having order k, for n >= 1, 1 <= k <= A051593(n).

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A303728 Triangle read by rows: T(n,k) is the number of labeled cyclic subgroups of order k in the alternating group A_n, 1 <= k <= A051593(n).

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A053039 Exponent of largest power of 2 which appears in the cototient-iteration started with n!.

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A181950 Weighted sum of all cyclic subgroups of the Alternating Group A_n.

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A057742 Maximal order of element of alternating group A_{2n}.

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A057743 Maximal order of element of alternating group A_{2n+1}.

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A355572 Largest LCM of partitions of n into odd parts.

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PARI

A355573 Largest LCM of partitions of n with a nonzero even number of even parts.

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