A227404 -id:A227404 - cp's OEIS frontend

A227413 a(1)=1, a(2n)=nthprime(a(n)), a(2n+1)=nthcomposite(a(n)), where nthprime = A000040, nthcomposite = A002808.

1, 2, 4, 3, 6, 7, 9, 5, 8, 13, 12, 17, 14, 23, 16, 11, 10, 19, 15, 41, 22, 37, 21, 59, 27, 43, 24, 83, 35, 53, 26, 31, 20, 29, 18, 67, 30, 47, 25, 179, 58, 79, 34, 157, 54, 73, 33, 277, 82, 103, 40, 191, 62, 89, 36, 431, 114, 149, 51, 241, 75, 101, 39, 127, 46

Offset: 1

Antti Karttunen, Jul 10 2013

Inverse permutation of A135141.

Shares with A073846 the property that the other bisection consists of just primes and the other bisection of just nonprimes.

Antti Karttunen and Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for sequences that are permutations of the natural numbers

Similarly constructed permutations: A227402, A227404, A227410, A227412. Cf. also A073846, A209636.

import Data.List (transpose)
a227413 n = a227413_list !! (n-1)
a227413_list = 1 : concat (transpose [map a000040 a227413_list,
                                      map a002808 a227413_list])
-- Reinhard Zumkeller, Jan 29 2014

a(1)=1, a(2n) = A000040(a(n)), a(2n+1) = A002808(a(n)).

A007097(n) = a(A000079(n)).

A216239 Total number of inversions in all derangement permutations of [n].

Original entry on oeis.org

0, 0, 1, 4, 34, 260, 2275, 21784, 228676, 2614296, 32372805, 431971100, 6182204006, 94495208444, 1536740258599, 26498747241680, 482990781797000, 9279452377499504, 187442757190618761, 3971627425918503156, 88084356619901450410, 2040857112777615061300

Offset: 0

Alois P. Heinz, Mar 15 2013

nonn

			a(2) = 1: (2,1) has 1 inversion.
a(3) = 4: (2,3,1), (3,1,2) have 2+2 = 4 inversions.
a(4) = 34: (2,1,4,3), (2,3,4,1), (2,4,1,3), (3,1,4,2), (3,4,1,2), (3,4,2,1), (4,1,2,3), (4,3,1,2), (4,3,2,1) have 2+3+3+3+4+5+3+5+6 = 34 inversions.

Max Alekseyev and Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 100 terms from Max Alekseyev)
Moussa Ahmia, José L. Ramírez, and Diego Villamizar, Inversions in Colored Permutations, Derangements, and Involutions, arXiv:2505.01550 [math.CO], 2025. See p. 12.
Wikipedia, Derangement
Wikipedia, Inversion

Cf. A000166, A001809, A161124, A211606, A227404, A228924, A337193.

v:= proc(l) local i; for i to nops(l) do if l[i]=i then return 0 fi od;
      add(add(`if`(l[i]>l[j], 1, 0), j=i+1..nops(l)), i=1..nops(l)-1)
    end:
a:= n-> add(v(d), d=combinat[permute](n)):
seq(a(n), n=0..8);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, n*(n-1)/2,
      n*((6*n^3-26*n^2+31*n-9)*a(n-1)+(n-1)*
      (6*n^2-8*n+1)*a(n-2))/((n-2)*(15-20*n+6*n^2)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 13 2013

A216239[n_] := (1/12)*n*(3*(-1)^n*n + (n*(3*n - 1) + 1)*Subfactorial[n-1]); Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Feb 05 2015, after Max Alekseyev *)

A216239(n) = sum(k=0,n-2, (-1)^k * n!/k! * (3*n+k) * (n-k-1) )/12; /* Max Alekseyev, Aug 13 2013 */

a(n) = SUM(k=0..n-2, (-1)^k * n!/k! * (3*n+k)*(n-k-1) )/12. - Max Alekseyev, Aug 13 2013
a(n) = ( (3*n^2-n+1)*A000166(n) + (n-1)*(-1)^n )/12. - Max Alekseyev, Aug 14 2013
a(n) = Sum_{k>=1} A228924(n,k) * k. - Alois P. Heinz, Sep 22 2013
a(n) ~ n! * n^2 / (4*exp(1)). - Vaclav Kotesovec, Sep 10 2014

Formula and terms a(15) onward from Max Alekseyev, Aug 13 2013

cp's OEIS Frontend

A227413 a(1)=1, a(2n)=nthprime(a(n)), a(2n+1)=nthcomposite(a(n)), where nthprime = A000040, nthcomposite = A002808.

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A216239 Total number of inversions in all derangement permutations of [n].

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