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A122111 Self-inverse permutation of the positive integers induced by partition enumeration in A112798 and partition conjugation. 107

%I

%S 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 96,25,40,1024,30,2048,11,72,192,54,21,4096,384,144,28,8192,60,16384,

%U 80,50,768,32768,22,81,45,288,160,65536,35,108,56,576,1536,131072,42

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

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

%C From _Antti Karttunen_, May 12-19 2014: (Start)

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

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

%C 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 (End)

%C From _Antti Karttunen_, Jul 31 2014: (Start)

%C 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 Each even number is mapped to an unique term of A102750 and vice versa.

%C Conversely, each odd number (larger than 1) is mapped to an 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 Taking the odd bisection and dividing the largest prime factor out results the permutation A243505.

%C 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 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 (End)

%C 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 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 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 A. Karttunen, <a href="/A122111/a122111.txt">A few notes on A122111, A241909 & A241916.</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F From _Antti Karttunen_, May 12-19 2014: (Start)

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

%F and then using any of the following three alternative formulas:

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

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

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

%F (End)

%F From _Antti Karttunen_, Jul 31 2014: (Start)

%F As a composition of related permutations:

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

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

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

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

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

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

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

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

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

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

%F A238690(a(n)) = A238690(n). - per _Matthew Vandermast_'s note in that sequence.

%F A238745(n) = a(A181819(n)) and a(A238745(n)) = A181819(n). - per _Matthew Vandermast_'s note in A238745.

%F A181815(n) = a(A181820(n)) and a(A181815(n)) = A181820(n). - per _Matthew Vandermast_'s note in A181815.

%F (End)

%p 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 # second Maple program:

%p a:= n-> (l-> mul(ithprime(add(`if`(j<i, 0, 1), j=l)), i=1..max(l)))(

%p [seq(numtheory[pi](i[1])$i[2], i=ifactors(n)[2])]):

%p seq(a(n), n=1..60); # _Alois P. Heinz_, Sep 30 2017

%t 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 *)

%o (Scheme, three alternative definitions, with _Antti Karttunen_'s IntSeq-library)

%o (definec (A122111 n) (if (<= n 1) n (* (A000040 (A001222 n)) (A122111 (A064989 n)))))

%o (definec (A122111 n) (if (<= n 1) n (* (A000079 (A241917 n)) (A003961 (A122111 (A052126 n))))))

%o (definec (A122111 n) (if (<= n 1) n (* (expt (A000040 (A071178 n)) (A241919 n)) (A242378bi (A071178 n) (A122111 (A051119 n))))))

%o ;; _Antti Karttunen_, May 12 2014

%o (Python)

%o from sympy import factorint, prevprime, prime, primefactors

%o from operator import mul

%o def a001222(n): return 0 if n==1 else a001222(n/primefactors(n)[0]) + 1

%o def a064989(n):

%o f=factorint(n)

%o return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])

%o def a105560(n): return 1 if n==1 else prime(a001222(n))

%o def a(n): return 1 if n==1 else a105560(n)*a(a064989(n))

%o print [a(n) for n in xrange(1, 101)] # _Indranil Ghosh_, Jun 15 2017

%Y Cf. A088902 (fixed points).

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

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

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

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

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

%Y Other related permutations: A048673-A064216, A244981-A244982, A244983-A244984, A243287-A243288, A243505-A243506, A245613-A245614, A075157, A075158, A129594, A069799, A242415, A245451, A245452, A245454.

%K nonn,nice

%O 1,2

%A _Franklin T. Adams-Watters_, Oct 18 2006

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