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

%I A122111 #110 Feb 13 2025 17:36:48
%S A122111 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,
%T A122111 96,25,40,1024,30,2048,11,72,192,54,21,4096,384,144,28,8192,60,16384,
%U A122111 80,50,768,32768,22,81,45,288,160,65536,35,108,56,576,1536,131072,42
%N A122111 Self-inverse permutation of the positive integers induced by partition enumeration in A112798 and partition conjugation.
%C A122111 Factor n; replace each prime(i) with i, take the conjugate partition, replace parts i with prime(i) and multiply out.
%C A122111 From _Antti Karttunen_, May 12-19 2014: (Start)
%C A122111 For all n >= 1, A001222(a(n)) = A061395(n), and vice versa, A061395(a(n)) = A001222(n).
%C A122111 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.
%C A122111 (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).
%C A122111 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.)
%C A122111 (End)
%C A122111 From _Antti Karttunen_, Jul 31 2014: (Start)
%C A122111 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.
%C A122111 Each even number is mapped to a unique term of A102750 and vice versa.
%C A122111 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.
%C A122111 Taking the odd bisection and dividing out the largest prime factor results in the permutation A243505.
%C A122111 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.
%C A122111 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.
%C A122111 (End)
%C A122111 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
%C A122111 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
%H A122111 Alois P. Heinz, <a href="/A122111/b122111.txt">Table of n, a(n) for n = 1..10000</a> (first 1024 terms from Antti Karttunen)
%H A122111 Antti Karttunen, <a href="/A122111/a122111.txt">A few notes on A122111, A241909 & A241916.</a>
%H A122111 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A122111 From _Antti Karttunen_, May 12-19 2014: (Start)
%F A122111 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).
%F A122111 This can be implemented as a recurrence, with base case a(1) = 1,
%F A122111 and then using any of the following three alternative formulas:
%F A122111   a(n) = A105560(n) * a(A064989(n)) = A000040(A001222(n)) * a(A064989(n)). [Cf. the formula for A242424.]
%F A122111   a(n) = A000079(A241917(n)) * A003961(a(A052126(n))).
%F A122111   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.]
%F A122111 a(n) = 1 + A075157(A129594(A075158(n-1))). [Follows from the commutativity with A241909, please see the comments section.]
%F A122111 (End)
%F A122111 From _Antti Karttunen_, Jul 31 2014: (Start)
%F A122111 As a composition of related permutations:
%F A122111 a(n) = A153212(A242419(n)) = A242419(A153212(n)).
%F A122111 a(n) = A241909(A241916(n)) = A241916(A241909(n)).
%F A122111 a(n) = A243505(A048673(n)).
%F A122111 a(n) = A064216(A243506(n)).
%F A122111 Other identities. For all n >= 1, the following holds:
%F A122111 A006530(a(n)) = A105560(n). [The latter sequence gives greatest prime factor of the n-th term].
%F A122111 a(2n)/a(n) = A105560(2n)/A105560(n), which is equal to A003961(A105560(n))/A105560(n) when n > 1.
%F A122111 A243505(n) = A052126(a(2n-1)) = A052126(a(4n-2)).
%F A122111 A066829(n) = A244992(a(n)) and vice versa, A244992(n) = A066829(a(n)).
%F A122111 A243503(a(n)) = A243503(n). [Because partition conjugation does not change the partition size.]
%F A122111 A238690(a(n)) = A238690(n). - per _Matthew Vandermast_'s note in that sequence.
%F A122111 A238745(n) = a(A181819(n)) and a(A238745(n)) = A181819(n). - per _Matthew Vandermast_'s note in A238745.
%F A122111 A181815(n) = a(A181820(n)) and a(A181815(n)) = A181820(n). - per _Matthew Vandermast_'s note in A181815.
%F A122111 (End)
%F A122111 a(n) = A181819(A108951(n)). [Prime shadow of the primorial inflation of n] - _Antti Karttunen_, Apr 29 2022
%p A122111 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
%p A122111 # second Maple program:
%p A122111 a:= n-> (l-> mul(ithprime(add(`if`(j<i, 0, 1), j=l)), i=1..max(l)))(
%p A122111         [seq(numtheory[pi](i[1])$i[2], i=ifactors(n)[2])]):
%p A122111 seq(a(n), n=1..60);  # _Alois P. Heinz_, Sep 30 2017
%t A122111 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 *)
%t A122111 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]}]]];
%t A122111 Array[a, 60] (* _Jean-François Alcover_, Sep 23 2020, after _Alois P. Heinz_ *)
%o A122111 (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
%o A122111 (Scheme)
%o A122111 ;; Uses _Antti Karttunen_'s IntSeq-library.
%o A122111 (definec (A122111 n) (if (<= n 1) n (* (A000040 (A001222 n)) (A122111 (A064989 n)))))
%o A122111 ;; _Antti Karttunen_, May 12 2014
%o A122111 (Scheme)
%o A122111 ;; Uses _Antti Karttunen_'s IntSeq-library.
%o A122111 (definec (A122111 n) (if (<= n 1) n (* (A000079 (A241917 n)) (A003961 (A122111 (A052126 n))))))
%o A122111 ;; _Antti Karttunen_, May 12 2014
%o A122111 (Scheme)
%o A122111 ;; Uses _Antti Karttunen_'s IntSeq-library.
%o A122111 (definec (A122111 n) (if (<= n 1) n (* (expt (A000040 (A071178 n)) (A241919 n)) (A242378bi (A071178 n) (A122111 (A051119 n))))))
%o A122111 ;; _Antti Karttunen_, May 12 2014
%o A122111 (Python)
%o A122111 from sympy import factorint, prevprime, prime, primefactors
%o A122111 from operator import mul
%o A122111 def a001222(n): return 0 if n==1 else a001222(n/primefactors(n)[0]) + 1
%o A122111 def a064989(n):
%o A122111     f=factorint(n)
%o A122111     return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
%o A122111 def a105560(n): return 1 if n==1 else prime(a001222(n))
%o A122111 def a(n): return 1 if n==1 else a105560(n)*a(a064989(n))
%o A122111 [a(n) for n in range(1, 101)] # _Indranil Ghosh_, Jun 15 2017
%Y A122111 Cf. A088902 (fixed points).
%Y A122111 Cf. A112798, A241918 (conjugates the partitions listed in these two tables).
%Y A122111 Cf. A243060 and A243070. (Limit of rows in these arrays, and also their central diagonal).
%Y A122111 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.
%Y A122111 Cf. A319988 (parity of this sequence for n > 1), A336124 (a(n) mod 4).
%Y A122111 {A000027, A122111, A241909, A241916} form a 4-group.
%Y A122111 {A000027, A122111, A153212, A242419} form also a 4-group.
%Y A122111 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).
%Y A122111 Cf. also array A350066 [A(i, j) = a(a(i)*a(j))].
%K A122111 nonn,nice
%O A122111 1,2
%A A122111 _Franklin T. Adams-Watters_, Oct 18 2006