A246353 If n = Sum 2^e_i, e_i distinct, then a(n) = Position of (product prime_{e_i+1}) among squarefree numbers (A005117).

1, 2, 3, 5, 4, 7, 11, 19, 6, 10, 14, 28, 23, 44, 65, 129, 8, 15, 21, 41, 34, 69, 101, 203, 48, 94, 144, 283, 233, 470, 703, 1405, 9, 17, 26, 49, 40, 80, 120, 236, 57, 111, 168, 334, 279, 554, 833, 1661, 89, 176, 261, 521, 438, 873, 1304, 2610, 609, 1217, 1827, 3650, 3046, 6091, 9131

Offset

0,2

Comments

This is an inverse function to A048672. Note the indexing: here the domain starts from 0, but the range starts from 1, while in A048672 it is the opposite.

Sequence is obtained when the range of A019565 is compacted so that it becomes surjective on N, thus the logarithmic scatter plots look very similar. (Same applies to A064273). Compare also to the plot of A005940.

Links

Antti Karttunen, Table of n, a(n) for n = 0..478

Index entries for sequences that are permutations of the natural numbers

Formula

a(n) = A013928(1+A019565(n)) = 1 + A013928(A019565(n)).

a(n) = A064273(n) + 1.

For all n >= 0, A048672(a(n)) = n.

For all n >= 1, a(A048672(n)) = n.

Prog

(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]])~))}; \\ This function from M. F. Hasler
A246353(n) = 1+A013928(A019565(n));
for(n=0, 478, write("b246353.txt", n, " ", A246353(n)));
(Scheme) (definec (A246353 n) (let loop ((n n) (i 1) (p 1)) (cond ((zero? n) (A013928 (+ 1 p))) ((odd? n) (loop (/ (- n 1) 2) (+ 1 i) (* p (A000040 i)))) (else (loop (/ n 2) (+ 1 i) p)))))
(Python)
from math import prod, isqrt
from sympy import prime, mobius
def A246353(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))) # Chai Wah Wu, Feb 22 2025

Crossrefs

Cf. A000040, A005940, A013928, A019565, A048672, A064273.

Sequence in context: A259153 A355066 A028691 * A194009 A242388 A257985

Adjacent sequences: A246350 A246351 A246352 * A246354 A246355 A246356

Keyword

nonn

Author

Antti Karttunen, Aug 23 2014

Status

approved