A372540 Least k such that the k-th squarefree number has binary expansion of length n. Index of the smallest squarefree number >= 2^n.

1, 2, 4, 7, 12, 21, 40, 79, 158, 315, 625, 1246, 2492, 4983, 9963, 19921, 39845, 79689, 159361, 318726, 637462, 1274919, 2549835, 5099651, 10199302, 20398665, 40797328, 81594627, 163189198, 326378285, 652756723, 1305513584, 2611027095, 5222054082, 10444108052

Offset: 0

Author

Gus Wiseman, May 10 2024

Keywords

• nonn
• base

Examples

			The squarefree numbers A005117(a(n)) together with their binary expansions and binary indices begin:
       1:                  1 ~ {1}
       2:                 10 ~ {2}
       5:                101 ~ {1,3}
      10:               1010 ~ {2,4}
      17:              10001 ~ {1,5}
      33:             100001 ~ {1,6}
      65:            1000001 ~ {1,7}
     129:           10000001 ~ {1,8}
     257:          100000001 ~ {1,9}
     514:         1000000010 ~ {2,10}
    1027:        10000000011 ~ {1,2,11}
    2049:       100000000001 ~ {1,12}
    4097:      1000000000001 ~ {1,13}
    8193:     10000000000001 ~ {1,14}
   16385:    100000000000001 ~ {1,15}
   32770:   1000000000000010 ~ {2,16}
   65537:  10000000000000001 ~ {1,17}
  131073: 100000000000000001 ~ {1,18}
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..73

Crossrefs

Counting zeros instead of length gives A372473, firsts of A372472.

For prime instead of squarefree we have:

- zeros A372474, firsts of A035103

- ones A372517, firsts of A014499

- bits A372684, firsts of A035100

Positions of first appearances in A372475, run-lengths A077643.

For weight instead of length we have A372541, firsts of A372433.

Indices of the squarefree numbers listed by A372683.

A000120 counts ones in binary expansion (binary weight), zeros A080791.

A005117 lists squarefree numbers.

A030190 gives binary expansion, reversed A030308.

A070939 counts bits, binary length, or length of binary expansion.

Cf. A029931, A048793, A049093, A049094, A059015, A069010, A143658, A211997, A230877.

Programs

Mathematica

nn=1000;
ssnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[Max@@#]&];
dcs=IntegerLength[Select[Range[nn],SquareFreeQ],2];
Table[Position[dcs,i][[1,1]],{i,ssnm[dcs]}]

Python

from itertools import count
from math import isqrt
from sympy import mobius, factorint
def A372540(n): return next(sum(mobius(a)*(k//a**2) for a in range(1, isqrt(k)+1)) for k in count(1<Chai Wah Wu, May 12 2024

Formula

A005117(a(n)) = A372683(n).

a(n) = A143658(n)+1 for n > 1. - Chai Wah Wu, Aug 26 2024

Extensions

a(24)-a(34) from Chai Wah Wu, May 12 2024