A249817 Permutation of natural numbers: a(1) = 1, a(n) = A083221(A055396(n),A246277(n)).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 39, 34, 35, 36, 37, 38, 63, 40, 41, 42, 43, 44, 33, 46, 47, 48, 49, 50, 75, 52, 53, 54, 65, 56, 99, 58, 59, 60, 61, 62, 57, 64, 95, 66, 67, 68, 111, 70, 71, 72, 73, 74, 51, 76, 77, 78, 79, 80, 45, 82, 83, 84, 155, 86, 135

Offset: 1

Author

Antti Karttunen, Nov 06 2014

Keywords

• nonn

Comments

a(n) tells which number in square array A083221 (the sieve of Eratosthenes) is at the same position where n is in array A246278. As both arrays have even numbers as their topmost row and primes as their leftmost column, both sequences are among the fixed points of this permutation.

Equally: a(n) tells which number in array A083140 is at the same position where n is in the array A246279, as they are the transposes of above two arrays.

Links

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

Crossrefs

Inverse: A249818.

There are three different "deep" versions of this permutation, recursing on values of A055396(n) and/or A246277(n), namely: A250245, A250247 and A250249.

Other similar or related permutations: A249815.

Cf. A000040, A005843, A020639, A055396, A078898, A083140, A083221, A246277, A246278, A246279, A249821.

Differs from its inverse A249818 for the first time at n=33, where a(33) = 39, while A249818(33) = 45.

Programs

Mathematica

lim = 87; a083221 = Table[Take[Prime[n] Select[Range[Ceiling[lim/2]^2], GCD[# Prime@ n, Product[Prime@ i, {i, 1, n - 1}]] == 1 &], Ceiling[lim/2]], {n, Ceiling[lim/2]}]; a055396[n_] PrimePi[FactorInteger[n][[1, 1]]]; a246277[n_] := Which[n == 1, 0, EvenQ@ n, n/2, True, a246277[Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ Transpose@ FactorInteger@ n, Last@ Transpose@ FactorInteger@ n]]]; Table[a083221[[a055396@ n, a246277@ n]], {n, 2, lim}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A055396 and Yasutoshi Kohmoto at A083140 *)

Scheme

(define (A249817 n) (if (= 1 n) n (A083221bi (A055396 n) (A246277 n)))) ;; Code for A083221bi given in A083221
;; Alternative version:
(define (A249817 n) (if (= 1 n) n (A083221bi (A055396 n) (A249821bi (A055396 n) (A078898 n))))) ;; Code for A249821bi given in A249821.

Formula

a(1) = 1, a(n) = A083221(A055396(n), A246277(n)).

a(1) = 1, a(n) = A083221(A055396(n), A249821(A055396(n), A078898(n))).

As a composition of other permutations:

a(1) = 1, and for n > 1, a(n) = 1 + A249815(n-1).

Other identities. For all n >= 1:

a(A005843(n)) = A005843(n) and a(A000040(n)) = A000040(n). [Fixes even numbers and primes, among other numbers. Cf. comments above].

A020639(a(n)) = A020639(n) and A055396(a(n)) = A055396(n). [Preserves the smallest prime factor of n].