A048640 Binary encoding of the squarefree numbers, A005117.

1, 2, 4, 8, 6, 16, 10, 32, 64, 18, 12, 128, 256, 20, 34, 512, 66, 1024, 14, 2048, 36, 130, 24, 4096, 258, 68, 8192, 22, 16384, 514, 32768, 132, 65536, 40, 260, 1026, 131072, 262144, 2050, 72, 38, 524288, 516, 26, 1048576, 2097152, 4098, 48, 70, 4194304

Offset: 1

Author

Antti Karttunen, Jul 14 1999

Keywords

• nonn

Examples

			10 = 2*5 = p_1*p_3 -> 2^1+2^3 = 2+8 = 10.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A048639.

Programs

Mathematica

Total[2^PrimePi@ # &@ Map[First, FactorInteger@ #]] & /@ Select[Range@ 80, SquareFreeQ] (* Michael De Vlieger, Oct 01 2015 *)

PARI

lista(nn) = {for (n=1, nn, if (issquarefree(n), if (n==1, x = n, f = factor(n); x = sum(k=1, #f~, 2^primepi(f[k,1]))); print1(x, ", ");););} \\ Michel Marcus, Oct 01 2015

Python

from math import isqrt
from sympy import mobius, primepi, primefactors
def A048640(n):
    if n == 1: return 1
    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<Chai Wah Wu, Dec 23 2024

Formula

a(n) = 2^i1+2^i2+...+2^iz, where A005117(n) = p_i1*p_i2*p_i3*...*p_iz (p_i stands for the i-th prime, where the first prime is 2).