A122111 Self-inverse permutation of the positive integers induced by partition enumeration in A112798 and partition conjugation.

1, 2, 4, 3, 8, 6, 16, 5, 9, 12, 32, 10, 64, 24, 18, 7, 128, 15, 256, 20, 36, 48, 512, 14, 27, 96, 25, 40, 1024, 30, 2048, 11, 72, 192, 54, 21, 4096, 384, 144, 28, 8192, 60, 16384, 80, 50, 768, 32768, 22, 81, 45, 288, 160, 65536, 35, 108, 56, 576, 1536, 131072, 42

Offset

1,2

Comments

Factor n; replace each prime(i) with i, take the conjugate partition, replace parts i with prime(i) and multiply out.

From Antti Karttunen, May 12-19 2014: (Start)

For all n >= 1, A001222(a(n)) = A061395(n), and vice versa, A061395(a(n)) = A001222(n).

Because the partition conjugation doesn't change the partition's total sum, this permutation preserves A056239, i.e., A056239(a(n)) = A056239(n) for all n.

(Similarly, for all n, A001221(a(n)) = A001221(n), because the number of steps in the Ferrers/Young-diagram stays invariant under the conjugation. - Note added Apr 29 2022).

Because this permutation commutes with A241909, in other words, as a(A241909(n)) = A241909(a(n)) for all n, from which follows, because both permutations are self-inverse, that a(n) = A241909(a(A241909(n))), it means that this is also induced when partitions are conjugated in the partition enumeration system A241918. (Not only in A112798.)

(End)

From Antti Karttunen, Jul 31 2014: (Start)

Rows in arrays A243060 and A243070 converge towards this sequence, and also, assuming no surprises at the rate of that convergence, this sequence occurs also as the central diagonal of both.

Each even number is mapped to a unique term of A102750 and vice versa.

Conversely, each odd number (larger than 1) is mapped to a unique term of A070003, and vice versa. The permutation pair A243287-A243288 has the same property. This is also used to induce the permutations A244981-A244984.

Taking the odd bisection and dividing out the largest prime factor results in the permutation A243505.

Shares with A245613 the property that each term of A028260 is mapped to a unique term of A244990 and each term of A026424 is mapped to a unique term of A244991.

Conversely, with A245614 (the inverse of above), shares the property that each term of A244990 is mapped to a unique term of A028260 and each term of A244991 is mapped to a unique term of A026424.

(End)

The Maple program follows the steps described in the first comment. The subprogram C yields the conjugate partition of a given partition. - Emeric Deutsch, May 09 2015

The Heinz number of the partition that is conjugate to the partition with Heinz number n. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r). Example: a(3) = 4. Indeed, the partition with Heinz number 3 is [2]; its conjugate is [1,1] having Heinz number 4. - Emeric Deutsch, May 19 2015

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1024 terms from Antti Karttunen)

A. Karttunen, A few notes on A122111, A241909 & A241916.

Index entries for sequences that are permutations of the natural numbers

Formula

From Antti Karttunen, May 12-19 2014: (Start)

a(1) = 1, a(p_i) = 2^i, and for other cases, if n = p_i1 * p_i2 * p_i3 * ... * p_{k-1} * p_k, where p's are primes, not necessarily distinct, sorted into nondescending order so that i1 <= i2 <= i3 <= ... <= i_{k-1} <= ik, then a(n) = 2^(ik-i_{k-1}) * 3^(i_{k-1}-i_{k-2}) * ... * p_{i_{k-1}}^(i2-i1) * p_ik^(i1).

This can be implemented as a recurrence, with base case a(1) = 1,

and then using any of the following three alternative formulas:

a(n) = A105560(n) * a(A064989(n)) = A000040(A001222(n)) * a(A064989(n)). [Cf. the formula for A242424.]

a(n) = A000079(A241917(n)) * A003961(a(A052126(n))).

a(n) = (A000040(A071178(n))^A241919(n)) * A242378(A071178(n), a(A051119(n))). [Here ^ stands for the ordinary exponentiation, and the bivariate function A242378(k,n) changes each prime p(i) in the prime factorization of n to p(i+k), i.e., it's the result of A003961 iterated k times starting from n.]

a(n) = 1 + A075157(A129594(A075158(n-1))). [Follows from the commutativity with A241909, please see the comments section.]

(End)

From Antti Karttunen, Jul 31 2014: (Start)

As a composition of related permutations:

a(n) = A153212(A242419(n)) = A242419(A153212(n)).

a(n) = A241909(A241916(n)) = A241916(A241909(n)).

a(n) = A243505(A048673(n)).

a(n) = A064216(A243506(n)).

Other identities. For all n >= 1, the following holds:

A006530(a(n)) = A105560(n). [The latter sequence gives greatest prime factor of the n-th term].

a(2n)/a(n) = A105560(2n)/A105560(n), which is equal to A003961(A105560(n))/A105560(n) when n > 1.

A243505(n) = A052126(a(2n-1)) = A052126(a(4n-2)).

A066829(n) = A244992(a(n)) and vice versa, A244992(n) = A066829(a(n)).

A243503(a(n)) = A243503(n). [Because partition conjugation does not change the partition size.]

A238690(a(n)) = A238690(n). - per Matthew Vandermast's note in that sequence.

A238745(n) = a(A181819(n)) and a(A238745(n)) = A181819(n). - per Matthew Vandermast's note in A238745.

A181815(n) = a(A181820(n)) and a(A181815(n)) = A181820(n). - per Matthew Vandermast's note in A181815.

(End)

a(n) = A181819(A108951(n)). [Prime shadow of the primorial inflation of n] - Antti Karttunen, Apr 29 2022

Maple

with(numtheory): c := proc (n) local B, C: B := proc (n) local pf: pf := op(2, ifactors(n)): [seq(seq(pi(op(1, op(i, pf))), j = 1 .. op(2, op(i, pf))), i = 1 .. nops(pf))] end proc: C := proc (P) local a: a := proc (j) local c, i: c := 0; for i to nops(P) do if j <= P[i] then c := c+1 else  end if end do: c end proc: [seq(a(k), k = 1 .. max(P))] end proc: mul(ithprime(C(B(n))[q]), q = 1 .. nops(C(B(n)))) end proc: seq(c(n), n = 1 .. 59); # Emeric Deutsch, May 09 2015
# second Maple program:
a:= n-> (l-> mul(ithprime(add(`if`(j<i, 0, 1), j=l)), i=1..max(l)))(
        [seq(numtheory[pi](i[1])$i[2], i=ifactors(n)[2])]):
seq(a(n), n=1..60);  # Alois P. Heinz, Sep 30 2017

Mathematica

A122111[1] = 1; A122111[n_] := Module[{l = #, m = 0}, Times @@ Power @@@ Table[l -= m; l = DeleteCases[l, 0]; {Prime@Length@l, m = Min@l}, Length@Union@l]] &@Catenate[ConstantArray[PrimePi[#1], #2] & @@@ FactorInteger@n]; Array[A122111, 60] (* JungHwan Min, Aug 22 2016 *)
a[n_] := Function[l, Product[Prime[Sum[If[j<i, 0, 1], {j, l}]], {i, 1, Max[l]}]][Flatten[Table[Table[PrimePi[f[[1]]], {f[[2]]}], {f, FactorInteger[n]}]]];
Array[a, 60] (* Jean-François Alcover, Sep 23 2020, after Alois P. Heinz *)

Prog

(PARI) A122111(n) = if(1==n, n, my(f=factor(n), es=Vecrev(f[, 2]), is=concat(apply(primepi, Vecrev(f[, 1])), [0]), pri=0, m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m)); \\ Antti Karttunen, Jul 20 2020
(Scheme, three alternative definitions, with Antti Karttunen's IntSeq-library)
(definec (A122111 n) (if (<= n 1) n (* (A000040 (A001222 n)) (A122111 (A064989 n)))))
(definec (A122111 n) (if (<= n 1) n (* (A000079 (A241917 n)) (A003961 (A122111 (A052126 n))))))
(definec (A122111 n) (if (<= n 1) n (* (expt (A000040 (A071178 n)) (A241919 n)) (A242378bi (A071178 n) (A122111 (A051119 n))))))
;; Antti Karttunen, May 12 2014
(Python)
from sympy import factorint, prevprime, prime, primefactors
from operator import mul
def a001222(n): return 0 if n==1 else a001222(n/primefactors(n)[0]) + 1
def a064989(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
def a105560(n): return 1 if n==1 else prime(a001222(n))
def a(n): return 1 if n==1 else a105560(n)*a(a064989(n))
[a(n) for n in range(1, 101)] # Indranil Ghosh, Jun 15 2017

Crossrefs

Cf. A088902 (fixed points).

Cf. A112798, A241918 (conjugates the partitions listed in these two tables).

Cf. A243060 and A243070. (Limit of rows in these arrays, and also their central diagonal).

Cf. also A080576, A036036, A001221, A001222, A061395, A243503, A056239, A052126, A003961, A064989, A071178, A241917, A241919, A242378, A242424, A006530, A105560, A070003, A102750, A066829, A028260, A026424, A244990, A244991, A244992, A108951, A181815, A181819, A181820, A238745, A238690, A242421.

Cf. A319988 (parity of this sequence for n > 1), A336124 (a(n) mod 4).

{A000027, A122111, A241909, A241916} form a 4-group.

{A000027, A122111, A153212, A242419} form also a 4-group.

Other related permutations: A048673-A064216, A244981-A244982, A244983-A244984, A243287-A243288, A243505-A243506, A245613-A245614, A075157, A075158, A129594, A069799, A242415, A245451, A245452, A245454, also A336321 & A336322 (composed with A225546).

Cf. also array A350066 [A(i, j) = a(a(i)*a(j))].

Sequence in context: A243288 A279352 A279351 * A243053 A243052 A153212

Adjacent sequences: A122108 A122109 A122110 * A122112 A122113 A122114

Keyword

nonn,nice

Author

Franklin T. Adams-Watters, Oct 18 2006

Status

approved