A181815 - OEIS

A181815

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

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, 72, 30, 192, 14, 54, 80, 512, 144, 60, 384, 28, 108, 25, 160, 1024, 45, 288, 21, 81, 120, 768, 56, 216, 50, 320, 2048, 90, 576, 11, 42, 162, 240, 1536, 112, 432, 100, 640, 4096, 180, 1152

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OFFSET

1,2

COMMENTS

A permutation of the natural numbers.

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

From Antti Karttunen, Dec 28 2019: (Start)

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

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.

(End)

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Antti Karttunen, Entanglement Permutations, 2016-2017.

Index entries for sequences that are permutations of the natural numbers

FORMULA

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).

a(n) = A122111(A181820(n)). - Matthew Vandermast, May 21 2012

From Antti Karttunen, Dec 24-29 2019: (Start)

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

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

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

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

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

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

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

(End)

EXAMPLE

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.

MATHEMATICA

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

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 *)

PROG

(PARI) A181815(n) = A329900(A025487(n)); \\ Antti Karttunen, Dec 24 2019

CROSSREFS

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).

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

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

Sequence in context: A244981 A284571 A124833 * A324931 A168521 A244982

Adjacent sequences: A181812 A181813 A181814 * A181816 A181817 A181818

KEYWORD

nonn,look

AUTHOR

Matthew Vandermast, Nov 30 2010

STATUS

approved