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)
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.
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).
From Antti Karttunen, Dec 24-29 2019: (Start)
(End)
EXAMPLE
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
(Python)
from math import prod
from itertools import count
from functools import lru_cache
from sympy import prime, factorint, integer_log
from oeis_sequences.OEISsequences import bisection
def A181815(n):
@lru_cache(maxsize=None)
def g(x, m, j): return sum(g(x//(prime(m)**i), m-1, i) for i in range(j, integer_log(x, prime(m))[0]+1)) if m-1 else max(0, x.bit_length()-j)
def f(x):
c, p = n-1+x, 1
for k in count(1):
p *= prime(k)
if p>x:
break
c -= g(x, k, 1)
return c
fs = factorint(bisection(f, n, n))
ps = sorted(fs)
l = len(ps)
return prod(ps[i]**(fs[ps[i]]-(fs[ps[i+1]] if i<l-1 else 0)) for i in range(l)) # Chai Wah Wu, Apr 01 2026
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).
KEYWORD
nonn,look
AUTHOR
Matthew Vandermast, Nov 30 2010
STATUS
approved
