This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
For n = 4 there is 1 permutation of order 1, 9 permutations of order 2, 8 of order 3 and 6 of order 4, for a total of 67.
b:= proc(n, g) option remember; `if`(n=0, g, add((j-1)!
*b(n-j, ilcm(g, j))*binomial(n-1, j-1), j=1..n))
end:
a:= n-> b(n, 1):
seq(a(n), n=0..30); # Alois P. Heinz, Jul 11 2017
CoefficientList[Series[Sum[n Fold[#1+MoebiusMu[n/#2] Apply[Times, Exp[x^#/#]&/@Divisors[#2] ]&,0,Divisors[n]],{n,Max[Apply[LCM,Partitions[19],1]]}],{x,0,19}],x] Range[0,19]! (* Wouter Meeussen, Jun 16 2012 *)
a[ n_] := If[ n < 1, Boole[n == 0], 1 + Total @ Apply[LCM, Map[Length, First /@ PermutationCycles /@ Drop[Permutations @ Range @ n, 1], {2}], 1]]; (* Michael Somos, Aug 19 2018 *)
\\ Naive method -- sum over cycles directly
cycleDecomposition(v:vec)={
my(cyc=List(), flag=#v+1, n);
while((n=vecmin(v))<#v,
my(cur=List(), i, tmp);
while(v[i++]!=n,);
while(v[i] != flag,
listput(cur, tmp=v[i]);
v[i]=flag;
i=tmp
);
if(#cur>1, listput(cyc, Vec(cur))) \\ Omit length-1 cycles
);
Vec(cyc)
};
permutationOrder(v:vec)={
lcm(apply(length, cycleDecomposition(v)))
};
a(n)=sum(i=0,n!-1,permutationOrder(numtoperm(n,i)))
\\ Charles R Greathouse IV, Nov 06 2014
A060014(n) = { my(factn = n!, part, nb, i, j, res = 0); forpart(part = n, nb = 1; j = 1; for(i = 1, #part, if (i == #part || part[i + 1] != part[i], nb *= (i + 1 - j)! * part[i]^(i + 1 - j); j = i + 1)); res += (factn / nb) * lcm(Vec(part))); res; } \\ Jerome Raulin, Jul 11 2017 (much faster, O(A000041(n)) vs O(n!))
Set of orders of all degree 7 permutations is {1,2,3,4,5,6,7,10,12} so a(7)=1+2+3+4+5+6+7+10+12=50.
b:= proc(n, i) option remember; (p->`if`(i*n=0, 1,
add(b(n-p^j, i-1)*p^j, j=1..ilog[p](n))+
b(n, i-1)))(`if`(i=0, 0, ithprime(i)))
end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..50); # Alois P. Heinz, Jul 12 2017
b[n_, i_] := b[n, i] = Function [p, If[i*n == 0, 1, Sum[b[n-p^j, i-1]*p^j, {j, 1, Floor@Log[p, n]}] + b[n, i-1]]][If[i == 0, 0, Prime[i]]];
a[n_] := b[n, PrimePi[n]];
a /@ Range[0, 50] (* Jean-François Alcover, Mar 14 2021, after Alois P. Heinz *)
Set of orders of all degree 5 even permutations is {1,2,3,5} so a(5)=1+2+3+5=11.
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