A372541 Least k such that the k-th squarefree number has exactly n ones in its binary expansion.

1, 3, 6, 11, 20, 60, 78, 157, 314, 624, 1245, 3736, 4982, 9962, 19920, 39844, 79688, 239046, 318725, 956194, 1912371, 2549834, 5099650, 15298984, 20398664, 40797327, 81594626, 163189197, 326378284, 979135127, 1305513583, 2611027094, 5222054081, 10444108051

Offset

0,2

Links

Chai Wah Wu, Table of n, a(n) for n = 0..56

Wikipedia, Hamming weight.

Example

The squarefree numbers A005117(a(n)) together with their binary expansions and binary indices begin:
       1:                   1 ~ {1}
       3:                  11 ~ {1,2}
       7:                 111 ~ {1,2,3}
      15:                1111 ~ {1,2,3,4}
      31:               11111 ~ {1,2,3,4,5}
      95:             1011111 ~ {1,2,3,4,5,7}
     127:             1111111 ~ {1,2,3,4,5,6,7}
     255:            11111111 ~ {1,2,3,4,5,6,7,8}
     511:           111111111 ~ {1,2,3,4,5,6,7,8,9}
    1023:          1111111111 ~ {1,2,3,4,5,6,7,8,9,10}
    2047:         11111111111 ~ {1,2,3,4,5,6,7,8,9,10,11}
    6143:       1011111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,13}
    8191:       1111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13}
   16383:      11111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14}
   32767:     111111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
   65535:    1111111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}
  131071:   11111111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17}

Mathematica

nn=10000;
spnm[y_]:=Max@@NestWhile[Most, y, Union[#]!=Range[0, Max@@#]&];
dcs=DigitCount[Select[Range[nn], SquareFreeQ], 2, 1];
Table[Position[dcs, i][[1, 1]], {i, spnm[dcs-1]}]

Prog

(Python)
from math import isqrt
from itertools import count
from sympy import factorint, mobius
from sympy.utilities.iterables import multiset_permutations
def A372541(n):
    if n==0: return 1
    for l in count(n):
        m = 1<<l
        for d in multiset_permutations('0'*(l-n)+'1'*n):
            k = m+int('0'+''.join(d), 2)
            if max(factorint(k).values(), default=0)==1:
                return sum(mobius(a)*(k//a**2) for a in range(1, isqrt(k)+1)) # Chai Wah Wu, May 10 2024

Crossrefs

Positions of firsts appearances in A372433.

Counting zeros instead of ones gives A372473, firsts in A372472.

For prime instead of squarefree we have A372517, firsts of A014499.

Counting bits (length) gives A372540, firsts of A372475, runs A077643.

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

A005117 lists squarefree numbers.

A030190 gives binary expansion, reversed A030308.

A048793 lists positions of ones in reversed binary expansion, sum A029931.

A145037, A097110 count ones minus zeros, for primes A372516, A177796.

A371571 lists positions of zeros in binary expansion, sum A359359.

A371572 lists positions of ones in binary expansion, sum A230877.

A372515 lists positions of zeros in reversed binary expansion, sum A359400.

Cf. A023416, A049093, A049094, A069010, A070939, A071403, A211997, A280296, A372474.

Sequence in context: A094989 A052467 A006127 * A122106 A007707 A298798

Adjacent sequences: A372538 A372539 A372540 * A372542 A372544 A372545

Keyword

nonn,base

Author

Gus Wiseman, May 09 2024

Extensions

a(23)-a(33) from Chai Wah Wu, May 10 2024

Status

approved