A181815 - OEIS

%I #53 Jan 03 2020 16:20:26

%S 1,2,4,3,8,6,16,12,5,32,9,24,10,64,18,48,20,128,36,15,96,7,27,40,256,

%T 72,30,192,14,54,80,512,144,60,384,28,108,25,160,1024,45,288,21,81,

%U 120,768,56,216,50,320,2048,90,576,11,42,162,240,1536,112,432,100,640,4096,180,1152

%N a(n) = largest integer such that, when any of its divisors divides A025487(n), the quotient is a member of A025487.

%C A permutation of the natural numbers.

%C The number of divisors of a(n) equals the number of ordered factorizations of A025487(n) as A025487(j)*A025487(k). Cf. A182762.

%C From _Antti Karttunen_, Dec 28 2019: (Start)

%C Rearranges terms of A108951 into ascending order, as A108951(a(n)) = A025487(n).

%C The scatter plot looks like a curtain of fractal spray, which is typical for many of the so-called "entanglement permutations". Indeed, according to the terminology I use in my 2016-2017 paper, this sequence is obtained by entangling the complementary pair (A329898, A330683) with the complementary pair (A005843, A003961), which gives the following implicit recurrence: a(A329898(n)) = 2*a(n) and a(A330683(n)) = A003961(a(n)). An explicit form is given in the formula section.

%C (End)

%H Amiram Eldar, <a href="/A181815/b181815.txt">Table of n, a(n) for n = 1..10000</a>

%H Antti Karttunen, <a href="/A135141/a135141.pdf">Entanglement Permutations</a>, 2016-2017.

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F If A025487(n) is considered in its form as Product A002110(i)^e(i), then a(n) = Product p(i)^e(i). If A025487(n) is instead considered as Product p(i)^e(i), then a(n) = Product (p(i)/A008578(i))^e(i).

%F a(n) = A122111(A181820(n)). - _Matthew Vandermast_, May 21 2012

%F From _Antti Karttunen_, Dec 24-29 2019: (Start)

%F a(n) = Product_{i=1..A051282(n)} A000040(A304886(i)).

%F a(n) = A329900(A025487(n)) = A319626(A025487(n)).

%F a(n) = A163511(A329905(n)).

%F For n > 1, if A330682(n) = 1, then a(n) = A003961(a(A329904(n))), otherwise a(n) = 2*a(A329904(n)).

%F A252464(a(n)) = A329907(n).

%F A330690(a(n)) = A050378(n).

%F a(A306802(n)) = A329902(n).

%F (End)

%e For any divisor d of 9 (d = 1, 3, 9), 36/d (36, 12, 4) is a member of A025487. 9 is the largest number with this relationship to 36; therefore, since 36 = A025487(11), a(11) = 9.

%t (* First, load the program at A025487, then: *)

%t Map[If[OddQ@ #, 1, Times @@ Prime@ # &@ Rest@ NestWhile[Append[#1, {#3, Drop[#, -LengthWhile[Reverse@ #, # == 0 &]] &[#2 - PadRight[ConstantArray[1, #3], Length@ #2]]}] & @@ {#1, #2, LengthWhile[#2, # > 0 &]} & @@ {#, #[[-1, -1]]} &, {{0, TakeWhile[If[# == 1, {0}, Function[g, ReplacePart[Table[0, {PrimePi[g[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, g]]@ FactorInteger@ #], # > 0 &]}}, And[FreeQ[#[[-1, -1]], 0], Length[#[[-1, -1]] ] != 0] &][[All, 1]] ] &, Union@ Flatten@ f@ 6] (* _Michael De Vlieger_, Dec 28 2019 *)

%o (PARI) A181815(n) = A329900(A025487(n)); \\ _Antti Karttunen_, Dec 24 2019

%Y If this sequence is considered the "primorial deflation" of A025487(n) (see first formula), the primorial inflation of n is A108951(n), and the primorial inflation of A025487(n) is A181817(n).

%Y A181820(n) is another mapping from the members of A025487 to the positive integers.

%Y Cf. A003961, A051282, A108951, A163511, A304886, A319626, A329897, A329898, A329900, A329901 (inverse), A329904, A329905, A329907, A330682 (reduced modulo 2), A330683.

%K nonn,look

%O 1,2

%A _Matthew Vandermast_, Nov 30 2010