A372683 Least squarefree number >= 2^n.

1, 2, 5, 10, 17, 33, 65, 129, 257, 514, 1027, 2049, 4097, 8193, 16385, 32770, 65537, 131073, 262145, 524289, 1048577, 2097154, 4194305, 8388609, 16777217, 33554433, 67108865, 134217730, 268435457, 536870913, 1073741826, 2147483649, 4294967297, 8589934594

Offset: 0

Author

Gus Wiseman, May 26 2024

Keywords

• nonn

Examples

			The terms 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}
  262145:  1000000000000000001 ~ {1,19}
  524289: 10000000000000000001 ~ {1,20}
		

Crossrefs

For primes instead of powers of two we have A112926, opposite A112925, sum A373197, length A373198.

Counting zeros instead of all bits gives A372473, firsts of A372472.

These are squarefree numbers at indices A372540, firsts of A372475.

Counting ones instead of all bits gives A372541, firsts of A372433.

The opposite (greatest squarefree number <= 2^n) is A372889.

The difference from 2^n is A373125.

For prime instead of squarefree we have:

- bits A372684, firsts of A035100

- zeros A372474, firsts of A035103

- ones A372517, firsts of A014499

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

A005117 lists squarefree numbers.

A030190 gives binary expansion, reversed A030308, length A070939 or A029837.

A061398 counts squarefree numbers between primes (exclusive).

A077643 counts squarefree terms between powers of 2, run-lengths of A372475.

A143658 counts squarefree numbers up to 2^n.

Cf. A029931, A048793, A049095, A049096, A059015, A069010, A076259, A077641, A211997, A230877.

Programs

Mathematica

Table[NestWhile[#+1&,2^n,!SquareFreeQ[#]&],{n,0,10}]

PARI

a(n) = my(k=2^n); while (!issquarefree(k), k++); k; \\ Michel Marcus, May 29 2024

Python

from itertools import count
from sympy import factorint
def A372683(n): return next(i for i in count(1<Chai Wah Wu, Aug 26 2024

Formula

a(n) = A005117(A372540(n)).

a(n) = A067535(2^n). - R. J. Mathar, May 31 2024