A182762 -id:A182762 - cp's OEIS frontend

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

Offset: 1

Matthew Vandermast, Nov 30 2010

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)

			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.

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

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.

(* 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 *)

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

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)

A182763 Numbers that set records for number of ordered factorizations as A025487(j)*A025487(k).

Original entry on oeis.org

1, 2, 4, 8, 16, 24, 48, 96, 192, 384, 576, 768, 1152, 2304, 4608, 9216, 11520, 17280, 23040, 34560, 69120, 138240, 276480, 414720, 552960, 829440, 1105920, 1658880, 3317760, 6635520, 9953280, 13271040, 19906560, 26542080, 39813120, 53084160, 79626240

Offset: 1

Matthew Vandermast, Nov 30 2010

nonn

Subsequence of A025487: it includes A025487(n) iff A182762(n) > A182762(m) for all m < n.

			24 has 6 factorizations into two members of A025487 (24 = 1*24 = 2*12 = 4*6 = 6*4 = 12*2 = 24*1). No smaller number has more than 5 such factorizations. Hence, 24 belongs to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..344
Amiram Eldar, Table of n, a(n) = A025487(i), i, A182762(i) = A182764(n) for n = 1..344

Subsequence of A181818.

Cf. A025487, A181815, A182762, A182764.

More terms from Max Alekseyev, Jan 22 2012

A182764 Number of ordered factorizations of A182763(n) as A025487(j)*A025487(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 21, 24, 27, 28, 30, 32, 36, 42, 48, 54, 56, 60, 64, 66, 72, 80, 88, 90, 96, 100, 104, 110, 112, 120, 130, 132, 135, 140, 144, 150, 160, 165, 176, 180, 192, 200, 208, 220, 224, 240, 260, 264, 270, 280, 288, 300

Offset: 1

Matthew Vandermast, Dec 01 2010

nonn

			24 has 6 factorizations into two members of A025487 (24 = 1*24 = 2*12 = 4*6 = 6*4 = 12*2 = 24*1). Since 24 = A182763(6), a(6) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..344

Cf. A025487, A182762, A182763.

a(n) = A182762(m) where m is such that A182763(n) = A025487(m).

More terms from Max Alekseyev, Jan 22 2012

cp's OEIS Frontend

A181815 a(n) = largest integer such that, when any of its divisors divides A025487(n), the quotient is a member of A025487.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A182763 Numbers that set records for number of ordered factorizations as A025487(j)*A025487(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A182764 Number of ordered factorizations of A182763(n) as A025487(j)*A025487(k).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions