A064273 Permutation of nonnegative integers: a(n) = A013928(A019565(n)).

0, 1, 2, 4, 3, 6, 10, 18, 5, 9, 13, 27, 22, 43, 64, 128, 7, 14, 20, 40, 33, 68, 100, 202, 47, 93, 143, 282, 232, 469, 702, 1404, 8, 16, 25, 48, 39, 79, 119, 235, 56, 110, 167, 333, 278, 553, 832, 1660, 88, 175, 260, 520, 437, 872, 1303, 2609, 608, 1216, 1826, 3649

Offset: 0

Author

Howard A. Landman, Sep 23 2001

Keywords

• easy
• nonn

Comments

From Antti Karttunen, Aug 24 2014: (Start)

The original name of the sequence was: "Inverse of sequence A048672 considered as a permutation of the nonnegative integers".

However, the real inverse to A048672 is A246353(n) (= a(n)+1), satisfying A246353(A048672(n)) = n for all n. This sequence subtracts one from the terms of A246353 so as to obtain a permutation of nonnegative integers (bijection [0..] --> [0..]).

Sequence is obtained when the range of A019565 is compacted so that it becomes surjective, thus the logarithmic scatter plots look very similar. (Same applies to A246353.) Compare also to the plot of A005940.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 0..478
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

One less than A246353.

Cf. A000040, A005940, A013928, A019565, A048672.

Programs

PARI

allocatemem(234567890);
default(primelimit, 2^22)
uplim_for_13928 = 13123111;
v013928 = vector(uplim_for_13928); A013928(n) = v013928[n];
v013928[1]=0; n=1; while((n < uplim_for_13928), if(issquarefree(n), v013928[n+1] = v013928[n]+1, v013928[n+1] = v013928[n]); n++);
A019565(n) = {factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ M. F. Hasler
A064273(n) = A013928(A019565(n));
for(n=0, 478, write("b064273.txt", n, " ", A064273(n))); \\ Antti Karttunen, Aug 23 2014

Python

from math import prod, isqrt
from sympy import prime, mobius
def A064273(n):
    m = prod(prime(i) for i,j in enumerate(bin(n)[-1:1:-1],1) if j=='1')
    return int(sum(mobius(k)*(m//k**2) for k in range(1, isqrt(m)+1))-1) # Chai Wah Wu, Feb 23 2025

Scheme

(define (A064273 n) (let loop ((n n) (i 1) (p 1)) (cond ((zero? n) (- (A013928 (+ 1 p)) 1)) ((odd? n) (loop (/ (- n 1) 2) (+ 1 i) (* p (A000040 i)))) (else (loop (/ n 2) (+ 1 i) p))))) ;; Antti Karttunen, Aug 23 2014

Formula

From Antti Karttunen, Aug 24 2014: (Start)

a(n) = A013928(A019565(n)).

a(n) = A246353(n) - 1.

(End)

Extensions

More terms from Carl R. White, Apr 19 2006

Name changed by Antti Karttunen, Aug 23 2014