A372683 - OEIS

%I #19 Aug 26 2024 02:10:26

%S 1,2,5,10,17,33,65,129,257,514,1027,2049,4097,8193,16385,32770,65537,

%T 131073,262145,524289,1048577,2097154,4194305,8388609,16777217,

%U 33554433,67108865,134217730,268435457,536870913,1073741826,2147483649,4294967297,8589934594

%N Least squarefree number >= 2^n.

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

%F a(n) = A067535(2^n). - _R. J. Mathar_, May 31 2024

%e The terms together with their binary expansions and binary indices begin:

%e        1:                    1 ~ {1}

%e        2:                   10 ~ {2}

%e        5:                  101 ~ {1,3}

%e       10:                 1010 ~ {2,4}

%e       17:                10001 ~ {1,5}

%e       33:               100001 ~ {1,6}

%e       65:              1000001 ~ {1,7}

%e      129:             10000001 ~ {1,8}

%e      257:            100000001 ~ {1,9}

%e      514:           1000000010 ~ {2,10}

%e     1027:          10000000011 ~ {1,2,11}

%e     2049:         100000000001 ~ {1,12}

%e     4097:        1000000000001 ~ {1,13}

%e     8193:       10000000000001 ~ {1,14}

%e    16385:      100000000000001 ~ {1,15}

%e    32770:     1000000000000010 ~ {2,16}

%e    65537:    10000000000000001 ~ {1,17}

%e   131073:   100000000000000001 ~ {1,18}

%e   262145:  1000000000000000001 ~ {1,19}

%e   524289: 10000000000000000001 ~ {1,20}

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

%o (PARI) a(n) = my(k=2^n); while (!issquarefree(k), k++); k; \\ _Michel Marcus_, May 29 2024

%o (Python)

%o from itertools import count

%o from sympy import factorint

%o def A372683(n): return next(i for i in count(1<<n) if max(factorint(i).values(),default=1)==1) # _Chai Wah Wu_, Aug 26 2024

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

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

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

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

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

%Y The difference from 2^n is A373125.

%Y For prime instead of squarefree we have:

%Y - bits A372684, firsts of A035100

%Y - zeros A372474, firsts of A035103

%Y - ones A372517, firsts of A014499

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

%Y A005117 lists squarefree numbers.

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

%Y A061398 counts squarefree numbers between primes (exclusive).

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

%Y A143658 counts squarefree numbers up to 2^n.

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

%K nonn

%O 0,2

%A _Gus Wiseman_, May 26 2024