A242420 -id:A242420 - cp's OEIS frontend

A243068 Fixed points of A242420.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 32, 36, 37, 41, 43, 47, 48, 49, 53, 54, 59, 61, 63, 64, 65, 67, 70, 71, 72, 73, 79, 81, 83, 89, 96, 97, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 133, 137, 139, 144

Offset: 1

Antti Karttunen, Jun 01 2014

nonn

A number n is present if its prime factorization n = p_a * p_b * p_c * ... * p_i * p_j * p_k^e_k, where a <= b <= c <= ... <= i <= j < k are the indices of prime factors, not necessarily all distinct, except that j < k, and the greatest prime divisor p_k [with k = A061395(n)] may occur multiple times, satisfies the condition that the first differences of those prime indices (a-0, b-a, c-b, ..., j-i, k-j) form a palindrome.

			4 = p_1^2 is present, as the first differences (deltas) of the prime indices (excluding the extra copies of the largest prime factor 2), form a palindrome: (1-0) = (1).
18 = 2*3*3 = p_1 * p_2 * p_2 is present, as the deltas of the indices of its nondistinct prime factors, (excluding the extra copies of the largest prime factor 3) form a palindrome: (1-0, 2-1) = (1,1).
60 = 2*2*3*5 = p_1 * p_1 * p_2 * p_3 is NOT present, as the deltas of prime indices (1-0, 1-1, 2-1, 3-2) = (1,0,1,1) do NOT form a palindrome.
Also, any of the cases mentioned in the Example section of A243058 as being present there, are also present in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..2185

Fixed points of A242420.
Differs from A242413 for the first time at n=36, where a(36)=61, while A242413(36)=60.
A000040 and A243058 are subsequences.
Cf. A242414, A242417.

A243286 Self-inverse permutation of natural numbers induced by the restriction of A243057 (or A243059 or A242420) to the union of {1} and A102750.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 12, 8, 9, 10, 27, 7, 13, 14, 36, 16, 62, 18, 19, 121, 148, 22, 23, 24, 43, 191, 11, 28, 283, 75, 113, 32, 87, 34, 481, 15, 388, 38, 39, 160, 1456, 42, 25, 795, 213, 602, 47, 74, 49, 818, 51, 52, 339, 54, 2699, 345, 57, 58, 59, 1053, 5219, 17, 914, 64

Offset: 1

Antti Karttunen, Jun 02 2014

nonn

			With n=7, the seventh number in the complement of A070003 (i.e. in the union of {1} and A102750) is A102750(6) = 10. When A243057 (or A243059) is applied to it, the result is another number that is a member of A102750, in this case 15, which occurs there as A102750(11). Thus a(7) = 11+1 = 12.

Antti Karttunen, Table of n, a(n) for n = 1..126
Index entries for sequences that are permutations of the natural numbers

Cf. A243057, A243059, A102750, A243285, A242420.

(define (A243286 n) (if (<= n 1) n (+ 1 (A243285 (A243057 (A102750 (- n 1)))))))

a(1) = 1, and for n>1, a(n) = 1 + A243285(A243057(A102750(n-1))). [Note: instead of A243057 one can also use A243059 or A242420.]

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)).

A243057 If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, with p_a <= p_b <= ... <= p_k, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 1 and for k>=1, p_{k} = A000040(k).

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 15, 11, 12, 13, 35, 10, 2, 17, 6, 19, 45, 21, 77, 23, 24, 5, 143, 3, 175, 29, 30, 31, 2, 55, 221, 14, 12, 37, 323, 91, 135, 41, 105, 43, 539, 20, 437, 47, 48, 7, 15, 187, 1573, 53, 6, 33, 875, 247, 667, 59, 90, 61, 899, 63, 2, 65, 385

Offset: 1

Antti Karttunen, May 31 2014

nonn

A243058 gives all n such that a(n) = n (the fixed points of this sequence, which include primes).
A102750 gives such n that a(a(n)) = n. A243286 is the self-inverse permutation induced when the domain is restricted to A102750. Cf. also A242420.
A070003 gives all n such that a(a(n)) <> n. Another variant, A243059, is defined to be zero when n is one of the terms of A070003.

			For n = 9 = 3*3 = p_2 * p_2, we have a(n) = p_{3-3} * p_3 = 1*3 = 3. [Like all terms in A070003 this is an example of "degenerate case", where some p's in the product get index 0, and thus are set to 1 by convention used here.]
For n = 10 = 2*5 = p_1 * p_3, we have a(n) = p_{3-1} * p_3 = 3*5 = 15.
For n = 12 = 2*2*3 = p_1 * p_1 * p_2, we have a(n) = p_{2-1} * p{2-1} * p_2 = p_1^2 * p_2 = 12.
For n = 15 = 3*5 = p_2 * p_3, we have a(n) = p_{3-2} * p_3 = 2*5 = 10.
For n = 2200 = 2*2*2*5*5*11 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5, we have a(n) = p_{5-3} * p_{5-3} * p_{5-1} * p_{5-1} * p_{5-1} * p_5 = 3*3*7*7*7*11 = 33957.
For n = 33957 = 3*3*7*7*7*11 = p_2 * p_2 * p_4 * p_4 * p_4 * p_5, we have a(n) = p_{5-4} * p_{5-4} * p_{5-4} * p_{5-2} * p_{5-2} * p_5 = 2*2*2*5*5*11 = 2200.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Fixed points: A243058 (includes primes).

Cf. A243059, A000040, A032742, A243056, A242419, A242420, A243286, A070003, A102750, A243074.

If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, where p_a <= p_b <= ... <= p_k are (not necessarily distinct) primes (sorted into nondescending order) in the prime factorization of n, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 1 and for k>=1, p_{k} = A000040(k).
a(1)=1, and for n>1, a(n) = p_{A243056(n)} * a(A032742(n)). Here p_{k} stands for 1 when k=0, and otherwise for the k-th prime, A000040(k).
For all n, a(n) = a(A243074(n)).
For all k in A102750, a(k) = A242420(k).

A243059 If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, with p_a <= p_b <= ... <= p_k, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 0 and for k>=1, p_{k} = A000040(k). a(1)=1 by convention.

Original entry on oeis.org

1, 2, 3, 0, 5, 6, 7, 0, 0, 15, 11, 12, 13, 35, 10, 0, 17, 0, 19, 45, 21, 77, 23, 24, 0, 143, 0, 175, 29, 30, 31, 0, 55, 221, 14, 0, 37, 323, 91, 135, 41, 105, 43, 539, 20, 437, 47, 48, 0, 0, 187, 1573, 53, 0, 33, 875, 247, 667, 59, 90, 61, 899, 63, 0, 65, 385, 67, 2873, 391, 70, 71, 0

Offset: 1

Antti Karttunen, May 31 2014

nonn

A243058 gives all n such that a(n) = n (the fixed points of this sequence, which include primes).

Differs from A243057 in that the "degenerate cases" A070003 are here zeros, but is otherwise equal to it (at the points given by A102750), i.e. for all n, a(A102750(n)) = A243057(A102750(n)) = A242420(A102750(n)).

			For n = 9 = 3*3 = p_2 * p_2, we have a(n) = p_{3-3} * p_3 = 0*3 = 0. [Like all terms in A070003 this is an example of "degenerate case", where some p's in the product get index 0, and thus are set to 0 by the convention used here.]
For n = 10 = 2*5 = p_1 * p_3, we have a(n) = p_{3-1} * p_3 = 3*5 = 15.
For n = 12 = 2*2*3 = p_1 * p_1 * p_2, we have a(n) = p_{2-1} * p{2-1} * p_2 = p_1^2 * p_2 = 12.
For n = 15 = 3*5 = p_2 * p_3, we have a(n) = p_{3-2} * p_3 = 2*5 = 10.
For n = 2200 = 2*2*2*5*5*11 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5, we have a(n) = p_{5-3} * p_{5-3} * p_{5-1} * p_{5-1} * p_{5-1} * p_5 = 3*3*7*7*7*11 = 33957.
For n = 33957 = 3*3*7*7*7*11 = p_2 * p_2 * p_4 * p_4 * p_4 * p_5, we have a(n) = p_{5-4} * p_{5-4} * p_{5-4} * p_{5-2} * p_{5-2} * p_5 = 2*2*2*5*5*11 = 2200.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Fixed points: A243058 (includes primes).
Positions of zeros: A070003.
Cf. A243057, A000040, A032742, A243056, A242419, A242420, A243286.

a(1)=1, and for n>1, a(n) = q_{A243056(n)} * a(A032742(n)). Here q_{k} stands for 0 when k=0, and otherwise for the k-th prime, A000040(k).

If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, where p_a <= p_b <= ... <= p_k are (not necessarily distinct) primes (sorted into nondescending order) in the prime factorization of n, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 0 and for k>=1, p_{k} = A000040(k).

cp's OEIS Frontend

A243068 Fixed points of A242420.

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A243286 Self-inverse permutation of natural numbers induced by the restriction of A243057 (or A243059 or A242420) to the union of {1} and A102750.

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

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A243057 If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, with p_a <= p_b <= ... <= p_k, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 1 and for k>=1, p_{k} = A000040(k).

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A243059 If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, with p_a <= p_b <= ... <= p_k, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 0 and for k>=1, p_{k} = A000040(k). a(1)=1 by convention.

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