A225891 -id:A225891 - cp's OEIS frontend

A069799 The number obtained by reversing the sequence of nonzero exponents in the prime factorization of n with respect to distinct primes present, as ordered by their indices.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 13, 14, 15, 16, 17, 12, 19, 50, 21, 22, 23, 54, 25, 26, 27, 98, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 250, 41, 42, 43, 242, 75, 46, 47, 162, 49, 20, 51, 338, 53, 24, 55, 686, 57, 58, 59, 150, 61, 62, 147, 64, 65

Offset: 1

Amarnath Murthy, Apr 13 2002

nonn

Equivalent description nearer to the old name: a(n) is a number obtained by reversing the indices of the primes present in the prime factorization of n, from the smallest to the largest, while keeping the nonzero exponents of those same primes at their old positions.
This self-inverse permutation of natural numbers fixes the numbers in whose prime factorization the sequence of nonzero exponents form a palindrome: A242414.
Integers which are changed are A242416.
Considered as a function on partitions encoded by the indices of primes in the prime factorization of n (as in table A112798), this implements an operation which reverses the order of vertical line segments of the "steps" in Young (or Ferrers) diagram of a partition, but keeps the order of horizontal line segments intact. Please see the last example in the example section.

			a(24) = 54 as 24 = p_1^3 * p_2^1 = 2^3 * 3^1 and 54 = p_1^1 * p_2^3 = 2 * 3^3.
For n = 2200, we see that it encodes the partition (1,1,1,3,3,5) in A112798 as 2200 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5 = 2^3 * 5^2 * 11. This in turn corresponds to the following Young diagram in French notation:
   _
  | |
  | |
  | |_ _
  |     |
  |     |_ _
  |_ _ _ _ _|
Reversing the order of vertical line segment lengths (3,2,1)  to (1,2,3), but keeping the order of horizontal line segment lengths as (1,2,2), we get a new Young diagram
   _
  | |_ _
  |     |
  |     |_ _
  |         |
  |         |
  |_ _ _ _ _|
which represents the partition (1,3,3,5,5,5), encoded in A112798 by p_1 * p_3^2 * p_5^3 = 2 * 5^2 * 11^3 = 66550, thus a(2200) = 66550.

Alois P. Heinz (first 1000 terms) & Antti Karttunen, Table of n, a(n) for n = 1..8192
Wikipedia, Young diagram
Index entries for sequences that are permutations of the natural numbers

A242414 gives the fixed points and A242416 is their complement.
Cf. A112798, A059404, A122111, A153212, A137502, A241916, A242418.
{A000027, A069799, A242415, A242419} form a 4-group.
The set of permutations {A069799, A105119, A225891} generates an infinite dihedral group.

a069799 n = product $
            zipWith (^) (a027748_row n) (reverse $ a124010_row n)
-- Reinhard Zumkeller, Apr 27 2013
(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define (A069799 n) (let ((pf (ifactor n))) (apply * (map expt (uniq pf) (reverse (multiplicities pf))))))
(define (ifactor n) (cond ((< n 2) (list)) (else (sort (factor n) <))))
(define (uniq lista) (let loop ((lista lista) (z (list))) (cond ((null? lista) (reverse! z)) ((and (pair? z) (equal? (car z) (car lista))) (loop (cdr lista) z)) (else (loop (cdr lista) (cons (car lista) z))))))
(define (multiplicities lista) (let loop ((mults (list)) (lista lista) (prev #f)) (cond ((not (pair? lista)) (reverse! mults)) ((equal? (car lista) prev) (set-car! mults (+ 1 (car mults))) (loop mults (cdr lista) prev)) (else (loop (cons 1 mults) (cdr lista) (car lista))))))
;; Antti Karttunen, May 24 2014

A069799 := proc(n) local e,j; e := ifactors(n)[2]:
mul (e[j][1]^e[nops(e)-j+1][2], j=1..nops(e)) end:
seq (A069799(i), i=1..40);
# Peter Luschny, Jan 17 2011

f[n_] := Block[{a = Transpose[ FactorInteger[n]], m = n}, If[ Length[a] == 2, Apply[ Times, a[[1]]^Reverse[a[[2]] ]], m]]; Table[ f[n], {n, 1, 65}]

a(n) = {my(f = factor(n)); my(g = f); my(nbf = #f~); for (i=1, nbf, g[i, 1] = f[nbf-i+1, 1];); factorback(g);} \\ Michel Marcus, Jul 02 2015

If n = p_a^e_a * p_b^e_b * ... * p_j^e_j * p_k^e_k, where p_a < ... < p_k are distinct primes of the prime factorization of n (sorted into ascending order), and e_a, ..., e_k are their nonzero exponents, then a(n) = p_a^e_k * p_b^e_j * ... * p_j^e_b * p_k^e_a.
a(n) = product(A027748(o(n)+1-k)^A124010(k): k=1..o(n)) = product(A027748(k)^A124010(o(n)+1-k): k=1..o(n)), where o(n) = A001221(n). - Reinhard Zumkeller, Apr 27 2013
From Antti Karttunen, Jun 01 2014: (Start)
Can be obtained also by composing/conjugating related permutations:
a(n) = A242415(A242419(n)) = A242419(A242415(n)).
a(n) = A122111(A242415(A122111(n))) = A153212(A242415(A153212(n))).
(End)

Edited, corrected and extended by Robert G. Wilson v and Vladeta Jovovic, Apr 15 2002
Definition corrected by Reinhard Zumkeller, Apr 27 2013
Definition again reworded, Comments section edited and Young diagram examples added by Antti Karttunen, May 30 2014

A242415 Reverse the deltas of indices of distinct primes in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 11, 12, 13, 35, 10, 16, 17, 18, 19, 45, 21, 77, 23, 24, 25, 143, 27, 175, 29, 30, 31, 32, 55, 221, 14, 36, 37, 323, 91, 135, 41, 105, 43, 539, 20, 437, 47, 48, 49, 75, 187, 1573, 53, 54, 33, 875, 247, 667, 59, 60, 61, 899, 63, 64, 65

Offset: 1

Antti Karttunen, May 24 2014

nonn

This self-inverse permutation (involution) of natural numbers preserves both the total number of prime divisors and the (index of) largest prime factor of n, i.e., for all n it holds that A001222(a(n)) = A001222(n) and A006530(a(n)) = A006530(n) [equally: A061395(a(n)) = A061395(n)]. It also preserves the exponent of the largest prime: A053585(a(n)) = A053585(n).
From the above it follows, that this fixes prime powers (A000961), among other numbers.
Considered as a function on partitions encoded by the indices of primes in the prime factorization of n (as in table A112798), this implements an operation which reverses the order of horizontal line segments of the "steps" in Young (or Ferrers) diagram of a partition, but keeps the order of vertical line segments intact. Please see the last example in the example section and compare also to the comments given in A242419.

			For n = 10 = 2*5 = p_1 * p_3, we get p_(3-1) * p_3 = 3 * 5 = 15, thus a(10) = 15.
For n = 20 = 2*2*5 = p_1^2 * p_3^1, we get p_(3-1)^2 * p_3^1 = 3^2 * 5 = 45, thus a(20) = 45.
For n = 84 = 2*2*3*7 = p_1^2 * p_2 * p_4, when we reverse the deltas of indices, but keep the exponents same, we get p_(4-2)^2 * p_(4-1) * p_4 = p_2^2 * p_3 * p_4 = 3^2 * 5 * 7 = 315, thus a(84) = 315.
For n = 2200, we see that it encodes the partition (1,1,1,3,3,5) in A112798 as 2200 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5 = 2^3 * 5^2 * 11. This in turn corresponds to the following Young diagram in French notation:
   _
  | |
  | |
  | |_ _
  |     |
  |     |_ _
  |_ _ _ _ _|
Reversing the order of horizontal line segment lengths (1,2,2) to (2,2,1), but keeping the order of vertical line segment lengths as (3,2,1), we get a new Young diagram
   _ _
  |   |
  |   |
  |   |_ _
  |       |
  |       |_
  |_ _ _ _ _|
which represents the partition (2,2,2,4,4,5), encoded in A112798 by p_2^3 * p_4^2 * p_5^1 = 3^3 * 7^2 * 11 = 14553, thus a(2200) = 14553.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Wikipedia, Young diagram
Index entries for sequences that are permutations of the natural numbers

Fixed points: A242413.
Cf. A112798, A122111, A153212, A000040, A241919, A067029, A242378, A051119, A225891.
{A000027, A069799, A242415, A242419} form a 4-group.

If n = p_a^e_a * p_b^e_b * ... * p_h^e_h * p_i^e_i * p_j^e_j * p_k^e_k, where p_a < ... < p_k are distinct primes (sorted into ascending order) in the prime factorization of n, and e_a .. e_k are their nonzero exponents, then a(n) = p_{k-j}^e_a * p_{k-i}^e_b  * p_{k-h}^e_c * ... * p_{k-a}^e_j * p_k^e_k.
As a recurrence: a(1) = 1, and for n>1, a(n) = (A000040(A241919(n))^A067029(n)) * A242378(A241919(n), a(A051119(A225891(n)))).
By composing/conjugating related permutations:
a(n) = A069799(A242419(n)) = A242419(A069799(n)).
a(n) = A122111(A069799(A122111(n))) = A153212(A069799(A153212(n))).

A105119 Numbers obtained by rotating right the indices in the prime signature of n.

Original entry on oeis.org

Offset: 1

Yasutoshi Kohmoto, Apr 08 2005

nonn

If n = p^a*q^b*r^c, then a(n) = p^c*q^a*r^b.
If n = p^a*q^b*r^c*s^d, then a(n) = p^d*q^a*r^b*s^c.
The sequence is a permutation of the positive integers. The first term which is different from A069799 is a(60).
Inverse permutation to A225891. The fixed points are A072774 (squarefree numbers and their powers). - Ivan Neretin, Jul 26 2015

			a(60) = a(2^2*3*5) = 2*3^2*5 = 90.

Permutation of A000027(n). Cf. A069799.

f:= proc(n) local F,j,m;
      F:= ifactors(n)[2];
      m:= nops(F);
      mul(F[i,1]^F[i-1,2],i=2..m)*F[1,1]^F[m,2] ;
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Jul 26 2015

Table[Times @@ ((ar = Transpose[FactorInteger[n]])[[1]]^RotateRight[ar[[2]]]), {n, 70}] (* Ivan Neretin, Jul 26 2015 *)

a(n)=local(m,s);m=factor(n);s=matsize(m)[1];prod(i=2,s,m[i,1]^m[i-1,2])*m[1,1]^m[s,2] /* Ralf Stephan, Apr 05 2009 */

Edited by Ralf Stephan, Apr 05 2009

a(1)=1 prepended by Ivan Neretin, Jul 26 2015

A242420 Self-inverse permutation of positive integers: a(n) = (A006530(n)^(A071178(n)-1)) * A243057(n).

Original entry on oeis.org

Offset: 1

Antti Karttunen, May 31 2014

nonn

This self-inverse permutation (involution) of positive integers preserves both the total number of prime divisors and the (index of) largest prime factor of n, i.e., for all n it holds that A001222(a(n)) = A001222(n) and A006530(a(n)) = A006530(n) [equally: A061395(a(n)) = A061395(n)].
It also preserves the exponent of the largest prime factor (A071178), from which follows that the sequence A102750 is closed with respect to this permutation, i.e., for all n in A102750, a(n) is either same n or some other term of A102750.
Considered as an operation on partitions encoded by the indices of primes in the prime factorization of n (as in table A112798), this implements a self-inverse bijection which is a composition of the effects of A242419 and A225891. (Or equally: A105119 and A242419). For details, please see the respective Comments sections and/or Example section of this entry.

			For n = 2200, we see that it encodes the partition (1,1,1,3,3,5) in A112798 as 2200 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5 = 2^3 * 5^2 * 11. This in turn corresponds to the following Young diagram in French notation:
   _
  | |
  | |
  | |_ _
  |     |
  |     |_ _
  |_ _ _ _ _|
First we apply A242419, which reverses the order of "steps", so that each horizontal and vertical line segment centered around a "convex corner" moves as a whole, so that the first stair from the top (one unit wide and three units high) is moved to the last position, the second one (two units wide and two units high) stays in the middle, and the original bottom step (two units wide and one unit high) will be the new topmost step, thus we get the following Young diagram:
   _ _
  |   |_ _
  |       |
  |       |_
  |         |
  |         |
  |_ _ _ _ _|
which represents the partition (2,4,4,5,5,5), encoded in A112798 by p_2 * p_4^2 * p_5^3 = 3 * 7^2 * 11^3 = 195657.
Then we apply A225891, which rotates the exponents of distinct primes in the factorization of n one left, in this context the vertical line segments one step up, with the top-one going to the bottomost, and so we get:
   _ _
  |   |
  |   |_ _
  |       |
  |       |
  |       |_
  |_ _ _ _ _|
which represents the partition (2,2,4,4,4,5), encoded in A112798 by p_2^2 * p_4^3 * p_5 = 3^2 * 7^3 * 11 = 33957, thus a(2200) = 33957.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Wikipedia, Young diagram
Index entries for sequences that are permutations of the natural numbers

Fixed points: A243068.

Cf. A112798, A242419, A105119, A225891, A006530, A071178, A102750, A243057, A243059.

(define (A242420 n) (* (expt (A006530 n) (- (A071178 n) 1)) (A243057 n)))

a(n) = (A006530(n)^(A071178(n)-1)) * A243057(n).
For all k in A102750, a(k) = A243057(k) = A243059(k).
By composing related permutations:
a(n) = A225891(A242419(n)) = A242419(A105119(n)).

cp's OEIS Frontend

A069799 The number obtained by reversing the sequence of nonzero exponents in the prime factorization of n with respect to distinct primes present, as ordered by their indices.

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A242415 Reverse the deltas of indices of distinct primes in the prime factorization of n.

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A105119 Numbers obtained by rotating right the indices in the prime signature of n.

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A242420 Self-inverse permutation of positive integers: a(n) = (A006530(n)^(A071178(n)-1)) * A243057(n).

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