A105119 -id:A105119 - 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

A242419 Reverse both the exponents and the deltas of the indices of distinct primes present in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 11, 18, 13, 35, 10, 16, 17, 12, 19, 75, 21, 77, 23, 54, 25, 143, 27, 245, 29, 30, 31, 32, 55, 221, 14, 36, 37, 323, 91, 375, 41, 105, 43, 847, 50, 437, 47, 162, 49, 45, 187, 1859, 53, 24, 33, 1715, 247, 667, 59, 150, 61, 899, 147, 64, 65

Offset: 1

Antti Karttunen, May 17 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)].
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 bijection which reverses the order of "steps" in Young (or Ferrers) diagram of a partition (but keeps the horizontal line segment of each step horizontal and the vertical line segment vertical). Please see the last example in the example section.
To understand the given recursive formula, it helps to see that in the above context (Young diagrams drawn with French notation), the sequences employed effect the following operations:
  A001222: gives the height of whole diagram,
  A051119: removes the bottommost step from the diagram,
  A241919: gives the length of the horizontal line segment of the bottom step, i.e. its width,
  A071178: gives the length of the vertical line segment of the bottom step, i.e. its height,
  A242378(k,n): increases the width of whole Young diagram encoded by n by adding a rectangular area A001222(n) squares high and k squares wide to its left,
  and finally, multiplying by A000040(a)^b adds a new topmost step whose width is a and height is b. Particularly, multiplying by (A000040(A241919(n))^A071178(n)) transfers the bottommost step to the top.

			For n = 10 = 2*5 = p_1^1 * p_3^1, we get p_(3-1)^1 * p_3^1 = 3 * 5 = 15, thus a(10) = 15.
For n = 20 = 2*2*5 = p_1^2 * p_3^1, we get p_(3-1)^1 * p_3^2 = 3^1 * 5^2 = 3*25 = 75, thus a(20) = 75.
For n = 84 = 2*2*3*7 = p_1^2 * p_2 * p_4, when we reverse the deltas of indices, and reverse also the order of exponents, we get p_(4-2) * p_(4-1) * p_4^2 = 3 * 5 * 7^2 = 735, thus a(84) = 735.
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 "steps", so that each horizontal and vertical line segment centered around a "convex corner" moves as a whole, means 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, thus a(2200) = 195657.

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: A242417.
Cf. A112798, A122111, A153212, A000040, A241919, A071178, A242378, A051119, A243057, A105119, A242420.
{A000027, A122111, A153212, A242419} form a 4-group.
{A000027, A069799, A242415, A242419} form also 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_k * p_{k-i}^e_j  * p_{k-h}^e_i * ... * p_{k-a}^e_b * p_k^e_a.
As a recurrence:
a(1) = 1, and for n>1, a(n) = (A000040(A241919(n))^A071178(n)) * A242378(A241919(n), a(A051119(n))).
By composing related permutations:
a(n) = A122111(A153212(n)) = A153212(A122111(n)).
a(n) = A069799(A242415(n)) = A242415(A069799(n)).
a(n) = A105119(A242420(n)).

A242420 Self-inverse permutation of positive integers: a(n) = (A006530(n)^(A071178(n)-1)) * A243057(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, 90, 61, 899, 63, 64, 65

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)).

A225891 Numbers obtained by rotating left the indices in the prime signature of n.

Original entry on oeis.org

Offset: 1

Paul Tek, May 20 2013

nonn

Inverse permutation to A105119.
A072774 gives the fixed points.
The first term which is different from A105119 is a(60).
The first term which is different from A069799 is a(90).

			a(2250) = a(2*3^2*5^3) = 2^2*3^3*5 = 540.

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A105119, A072774, A069799.

Table[Times @@ ((ar = Transpose[FactorInteger[n]])[[1]]^RotateLeft[ar[[2]]]), {n, 66}] (* Ivan Neretin, Jul 09 2015 *)

{ a(n)=my(f=factor(n),l=#f[,1]);
return(prod(i=1,l,f[i,1]^f[if(i==l,1,i+1),2])) }

A367167 Let Product_{i = 1..omega(n)} p_i^e_i be the prime factorization of n. Then a(n) = Sum_{i = 1..omega(n)} Product_{j = 1..n} p_i^(e_((i+j-1) mod omega(n) + 1)).

Original entry on oeis.org

0, 2, 3, 4, 5, 12, 7, 8, 9, 20, 11, 30, 13, 28, 30, 16, 17, 30, 19, 70, 42, 44, 23, 78, 25, 52, 27, 126, 29, 90, 31, 32, 66, 68, 70, 72, 37, 76, 78, 290, 41, 126, 43, 286, 120, 92, 47, 210, 49, 70, 102, 390, 53, 78, 110, 742, 114, 116, 59, 300, 61, 124, 210, 64, 130

Offset: 1

David James Sycamore and David A. Corneth Nov 07 2023

nonn

In other words, a(n) is constructed by incrementally shifting the elements of the ordered exponent multiset {e_1,e_2,...,e_k} to the right one place at a time, a total of k = A001221(n) times, and adding the results of these exponents applied to the distinct size ordered primes p_1,p_2,...,p_k which divide n. The first shift gives {e_k,e_1,e_2,...,e_(k-1)}, then {e_(k-1),e_k,...,e_(k-2)} ... and the last yields {e_1,e_2,...,e_k} (which is n itself). At each shift the last exponent of the previous shift becomes the first of the next. There are A001221(n) summands in the computation of a(n), each having the same rad (A007947) and omega (A001221) values as n.

a(n) >= n with equality for n > 1 in A000961.

			a(1) = 0, the empty sum.
a(6) = a(2^1*3^1) = 6 + 2^1*3^1 = 6 + 6 = 12.
a(12) = a(2^2*3^1) = 12 + 2^1*3^2 = 12 + 18 = 30.
a(2250) = a(2^1*3^2*5^3) = 2^3*3^1*5^2 + 2^2*3^3*5^1 + 2^1*3^2*5^3 = 3390.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Cf. A000961, A001221, A001222, A005117, A006881, A007947, A105119, A367140.

{0}~Join~Array[Total@ Table[Times @@ Power @@@ Transpose@ {#1, RotateRight[#2, k]}, {k, PrimeNu[#]}] & @@ Transpose@ FactorInteger[#] &, 64, 2] (* Michael De Vlieger, Nov 09 2023 *)

a(n) = {
	my(f = factor(n), res = 0);
	for(i = 1, #f~,
		res+=prod(j = 1, #f~, f[j, 1]^f[(i+j-1)%#f~ + 1, 2])
	);
	res
} \\ David A. Corneth, Nov 07 2023

a(n) = n for n > 1 in A000961.
a(n) = 2*n for n in A006881, and more generally if A007947(n) = n, a(n) = A001221(n)*n.
Let b(0) = n and let b(n+1) = A105119(b(n)) for n >= 0 and let omega(n) be the number of distinct prime factors of n. Then a(n) = Sum_{i = 1..omega(n)} b(i). - David A. Corneth, Nov 07 2023

More terms and revised name from David A. Corneth, Nov 07 2023

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A242419 Reverse both the exponents and the deltas of the indices of distinct primes present in the prime factorization of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula

A225891 Numbers obtained by rotating left the indices in the prime signature of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A367167 Let Product_{i = 1..omega(n)} p_i^e_i be the prime factorization of n. Then a(n) = Sum_{i = 1..omega(n)} Product_{j = 1..n} p_i^(e_((i+j-1) mod omega(n) + 1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions