A250475 -id:A250475 - cp's OEIS frontend

A251721 Square array of permutations: A(row,col) = A249822(row, A249821(row+1, col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

1, 2, 1, 3, 2, 1, 5, 3, 2, 1, 4, 4, 3, 2, 1, 7, 6, 4, 3, 2, 1, 11, 7, 5, 4, 3, 2, 1, 6, 9, 6, 5, 4, 3, 2, 1, 13, 10, 7, 6, 5, 4, 3, 2, 1, 17, 5, 8, 7, 6, 5, 4, 3, 2, 1, 10, 12, 10, 8, 7, 6, 5, 4, 3, 2, 1, 19, 15, 11, 9, 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, 13, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 8, 16, 14, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 19, 15, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Antti Karttunen, Dec 07 2014

These are the "first differences" between permutations of array A249821, in a sense that by composing the first k rows of this array [from left to right, as in a(n) = row_1(row_2(...(row_k(n))))], one obtains row k+1 of A249821.

On row n, the first A250473(n) terms are fixed, and the first non-fixed term comes at A250474(n).

			The top left corner of the array:
1, 2, 3, 5, 4, 7, 11, 6, 13, 17, 10, 19, 9, 8, 23, 29, 14, 15, 31, 22, ...
1, 2, 3, 4, 6, 7, 9, 10, 5, 12, 15, 8, 16, 19, 21, 22, 13, 24, 11, 27, ...
1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 9, 16, 18, 20, 21, 23, 24, ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, ...
...

Inverse permutations can be found from array A251722.
Row 1: A064216, Row 2: A249745, Row 3: A250475.
Cf. A000027, A002260, A004736, A078898, A083221, A246277, A246278, A249821, A249822, A250473, A250474.

(define (A251721bi row col) (A249822bi row (A249821bi (+ row 1) col)))
(define (A251721 n) (A251721bi (A002260 n) (A004736 n)))
;; Code for A249821bi and A249822bi given in A249821, A249822.

A(row,col) = A249822(row, A249821(row+1, col)).

A(row,col) = A078898(A246278(row, A246277(A083221(row+1, col)))).

A249824 Permutation of natural numbers: a(n) = A078898(A003961(A003961(2*n))).

Original entry on oeis.org

1, 2, 3, 9, 4, 12, 5, 42, 17, 19, 6, 59, 7, 22, 26, 209, 8, 82, 10, 92, 31, 29, 11, 292, 41, 32, 115, 109, 13, 129, 14, 1042, 40, 39, 48, 409, 15, 49, 45, 459, 16, 152, 18, 142, 180, 52, 20, 1459, 57, 202, 54, 159, 21, 572, 63, 542, 68, 62, 23, 642, 24, 69, 213

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

			a(4) = 9 because of the following. 2n = 2*4 = 8 = 2^3. We replace the prime factor 2 of 8 with the next prime 3 to get 3^3, then replace 3 with 5 to get 5^3 = 125. The smallest prime factor of 125 is 5. 125 is the 9th term of A084967: 5, 25, 35, 55, 65, 85, 95, 115, 125, ..., thus a(4) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..255
Index entries for sequences that are permutations of the natural numbers

Inverse: A249823.
Row 3 of A249822.
Cf. A003961, A048673, A078898, A084967, A243071, A246278, A249734, A249746, A249826, A250475, A275716.
Cf. also A273664, A273669.

t = PositionIndex[FactorInteger[#][[1, 1]] & /@ Range[10^4]]; f[n_] := Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; Flatten@ Table[Position[Lookup[t, FactorInteger[#][[1, 1]] ], #] &[f@ f[2 n]], {n, 120}] (* Michael De Vlieger, Jul 25 2016, Version 10 *)

(define (A249824 n) (A078898 (A003961 (A003961 (* 2 n)))))

a(n) = A078898(A003961(A003961(2*n))) = A078898(A003961(A249734(n))).
a(n) = A078898(A246278(3,n)).
As a composition of other permutations:
a(n) = A249746(A048673(n)).
a(n) = A250475(A249826(n)).
a(n) = A275716(A243071(n)).
Other identities. For all n >= 1:
a(2n) = A273669(a(n)) and a(A003961(n)) = A273664(a(n)). -- Antti Karttunen, Aug 07 2016

A249825 Permutation of natural numbers: a(n) = A246277(A084968(n)).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 4, 41, 43, 47, 53, 59, 61, 6, 67, 71, 73, 10, 79, 83, 89, 97, 101, 103, 14, 9, 107, 109, 22, 113, 127, 15, 131, 137, 139, 26, 149, 151, 25, 157, 163, 167, 21, 173, 179, 181, 191, 34, 33, 193, 38, 35, 197, 199, 211, 223, 227, 229, 55, 233, 39, 239, 46, 241, 251, 257, 263, 269, 271, 58, 49

Offset: 1

Antti Karttunen, Dec 06 2014

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

Inverse: A249826.
Row 4 of A249821.
Cf. A084968, A246277, A251721, A064216, A249823, A249745, A250475.

(define (A249825 n) (A246277 (A084968 n)))

a(n) = A246277(A084968(n)).
As a composition of other permutations:
a(n) = A249823(A250475(n)).
a(n) = A064216(A249745(A250475(n))). [Composition of the first three rows of array A251721.]

A250476 Permutation of natural numbers: a(n) = A249826(A249823(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 14, 9, 10, 21, 11, 12, 13, 15, 33, 16, 25, 17, 18, 32, 19, 20, 22, 39, 23, 24, 36, 26, 50, 43, 27, 28, 29, 30, 31, 34, 55, 56, 46, 92, 35, 37, 68, 38, 40, 59, 58, 41, 42, 70, 44, 86, 45, 47, 78, 48, 144, 49, 51, 77, 66, 52, 53, 54, 57, 91, 81, 80, 60, 88, 109, 87, 61, 62, 63, 64, 99, 65, 67, 226, 105, 69, 101, 71, 121, 72, 111, 73, 74

Offset: 1

Antti Karttunen, Dec 07 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A250475.
Row 3 of A251722.
Cf. A249823, A249826.

(define (A250476 n) (A249826 (A249823 n)))

a(n) = A249826(A249823(n)).

cp's OEIS Frontend

A251721 Square array of permutations: A(row,col) = A249822(row, A249821(row+1, col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A249824 Permutation of natural numbers: a(n) = A078898(A003961(A003961(2*n))).

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A249825 Permutation of natural numbers: a(n) = A246277(A084968(n)).

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A250476 Permutation of natural numbers: a(n) = A249826(A249823(n)).

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