OFFSET
1,3
COMMENTS
Permutation of nonnegative integers. Note the indexing, the domain starts from 1, although the range includes also 0.
A246353 gives the inverse of this sequence, in a sense that a(A246353(n)) = n for all n >= 0, and A246353(a(n)) = n for all n >= 1. When one is subtracted from the latter, another permutation of nonnegative integers is obtained: A064273. - Antti Karttunen, Aug 23 2014 based on comment from Howard A. Landman, Sep 25 2001
Also index of n-th term of A019565 when its terms are sorted in increasing order. For example: a(6) = 8. The smallest values of A019565 are 1,2,3,5,6,7 . The 6th is 7 which is A019565(8). - Philippe Lallouet (philip.lallouet(AT)orange.fr), Apr 28 2008
a(n) is the number whose binary indices are the prime indices of the n-th squarefree number (row n of A329631), where a binary index of n is any position of a 1 in its reversed binary expansion, and a prime index of n is a number m such that prime(m) divides n. The binary indices of n are row n of A048793, while the prime indices of n are row n of A112798. - Gus Wiseman, Nov 30 2019
LINKS
FORMULA
a(n) = 2^(i1-1)+2^(i2-1)+...+2^(iz-1), where A005117(n) = p_i1*p_i2*p_i3*...*p_iz.
EXAMPLE
From Gus Wiseman, Nov 30 2019: (Start)
The sequence of squarefree numbers together with their prime indices (A329631) and the number a(n) with those binary indices begins:
1 -> {} -> 0
2 -> {1} -> 1
3 -> {2} -> 2
5 -> {3} -> 4
6 -> {1,2} -> 3
7 -> {4} -> 8
10 -> {1,3} -> 5
11 -> {5} -> 16
13 -> {6} -> 32
14 -> {1,4} -> 9
15 -> {2,3} -> 6
17 -> {7} -> 64
19 -> {8} -> 128
21 -> {2,4} -> 10
22 -> {1,5} -> 17
23 -> {9} -> 256
26 -> {1,6} -> 33
29 -> {10} -> 512
30 -> {1,2,3} -> 7
(End)
MAPLE
encode_sqrfrees := proc(upto_n) local b, i; b := [ ]; for i from 1 to upto_n do if(0 <> mobius(i)) then b := [ op(b), bef(i) ]; fi; od: RETURN(b); end; # see A048623 for bef
MATHEMATICA
Join[{0}, Total[2^(PrimePi[FactorInteger[#][[All, 1]]] - 1)]& /@ Select[ Range[2, 100], SquareFreeQ]] (* Jean-François Alcover, Mar 15 2016 *)
PROG
(PARI) lista(nn) = {for (n=1, nn, if (issquarefree(n), if (n==1, x = 0, f = factor(n); x = sum(k=1, #f~, 2^(primepi(f[k, 1])-1))); print1(x, ", "); ); ); } \\ Michel Marcus, Oct 02 2015
(Python)
from math import isqrt
from sympy import mobius, primepi, primefactors
def A048672(n):
if n == 1: return 0
def f(x): return int(n-sum(mobius(k)*(x//k**2) for k in range(2, isqrt(x)+1)))
m, k = n, f(n)
while m != k: m, k = k, f(k)
return sum(1<<primepi(p)-1 for p in primefactors(m)) # Chai Wah Wu, Feb 22 2025
KEYWORD
easy,nonn
AUTHOR
Antti Karttunen, Jul 14 1999
STATUS
approved