A249818 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n),A078898(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, 45, 34, 35, 36, 37, 38, 33, 40, 41, 42, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 54, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 66, 67, 68, 135, 70, 71, 72, 73, 74, 51, 76, 77, 78, 79, 80, 99, 82, 83, 84, 175, 86, 105

Offset: 1

Author

Antti Karttunen, Nov 06 2014

Keywords

• nonn

Comments

a(n) tells which number in square array A246278 is at the same position where n is in array A083221, the sieve of Eratosthenes. 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 A246279 is at the same position where n is in the array A083140, 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: A249817.

There are three different "deep" versions of this permutation, recursing on values of A055396(n) and/or A078898(n), namely: A250246, A250248 and A250250.

Other similar or related permutations: A249816.

Cf. A000040, A005843, A020639, A055396, A078898, A083140, A083221, A246277, A246278, A246279, A249822.

Differs from its inverse A249817 for the first time at n=33, where a(33) = 45, while A249817(33) = 39.

Programs

Mathematica

lim = 87; a003961[p_?PrimeQ] := a003961[p] = Prime[PrimePi@ p + 1]; a003961[1] = 1; a003961[n_] :=  a003961[n] = Times @@ (a003961[First@ #]^Last@ # &) /@ FactorInteger@ n; a055396[n_] := PrimePi[FactorInteger[n][[1, 1]]]; a078898 = Block[{nn = 90, spfs}, spfs = Table[FactorInteger[n][[1, 1]], {n, nn}]; Table[Count[Take[spfs, i], spfs[[i]]], {i, nn}]]; a246278 = NestList[Map[a003961, #] &, Table[2 k, {k, lim}], lim]; Table[a246278[[a055396@ n, a078898[[n]]]], {n, 2, lim}]
(* Michael De Vlieger, Jan 04 2016, after Harvey P. Dale at A055396 and A078898 *)

Formula

a(1) = 1, a(n) = A246278(A055396(n), A078898(n)).

a(1) = 1, a(n) = A246278(A055396(n), A249822(A055396(n), A246277(n))).

As a composition of other permutations:

a(1) = 1, and for n > 1, a(n) = 1 + A249816(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].