A372473 - OEIS

A372473

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

1, 2, 7, 12, 21, 40, 79, 158, 315, 1247, 1246, 2492, 4983, 9963, 19921, 39845, 79689, 159361, 318726, 637462, 1274919, 2549835, 5099651, 10199302, 20398665, 40797328, 81594627, 163189198, 326378285, 652756723, 1305513584, 2611027095, 5222054082, 10444108052

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Note that the data is not strictly increasing.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..57

EXAMPLE

The squarefree numbers A005117(a(n)) together with their binary expansions and binary indices begin:

1: 1 ~ {1}

2: 10 ~ {2}

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}

2051: 100000000011 ~ {1,2,12}

2049: 100000000001 ~ {1,12}

4097: 1000000000001 ~ {1,13}

8193: 10000000000001 ~ {1,14}

MATHEMATICA

nn=10000;

spnm[y_]:=Max@@NestWhile[Most, y, Union[#]!=Range[0, Max@@#]&];

dcs=DigitCount[Select[Range[nn], SquareFreeQ], 2, 0];

Table[Position[dcs, i][[1, 1]], {i, 0, spnm[dcs]}]

PROG

(Python)

from math import isqrt

from itertools import count

from sympy import factorint, mobius

from sympy.utilities.iterables import multiset_permutations

def A372473(n):

if n==0: return 1

for l in count(n):

m = 1<<l

for d in multiset_permutations('0'*n+'1'*(l-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 first appearances in A372472.

For prime instead of squarefree we have A372474, A035103, A372517, A014499.

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

Counting 1's (weight) instead of 0's gives A372541, firsts of A372433.

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.

A070939 gives length of binary expansion (number of bits).

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. A035100, A039004, A049093, A049094, A059015, A069010, A145037, A211997.

Sequence in context: A119713 A213041 A293330 * A135541 A288656 A180804

Adjacent sequences: A372470 A372471 A372472 * A372474 A372475 A372476

KEYWORD

nonn,base

AUTHOR

Gus Wiseman, May 09 2024

EXTENSIONS

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

STATUS

approved