cp's OEIS Frontend

A372683 Least squarefree number >= 2^n.

%I A372683 #19 Aug 26 2024 02:10:26
%S A372683 1,2,5,10,17,33,65,129,257,514,1027,2049,4097,8193,16385,32770,65537,
%T A372683 131073,262145,524289,1048577,2097154,4194305,8388609,16777217,
%U A372683 33554433,67108865,134217730,268435457,536870913,1073741826,2147483649,4294967297,8589934594
%N A372683 Least squarefree number >= 2^n.
%F A372683 a(n) = A005117(A372540(n)).
%F A372683 a(n) = A067535(2^n). - _R. J. Mathar_, May 31 2024
%e A372683 The terms together with their binary expansions and binary indices begin:
%e A372683        1:                    1 ~ {1}
%e A372683        2:                   10 ~ {2}
%e A372683        5:                  101 ~ {1,3}
%e A372683       10:                 1010 ~ {2,4}
%e A372683       17:                10001 ~ {1,5}
%e A372683       33:               100001 ~ {1,6}
%e A372683       65:              1000001 ~ {1,7}
%e A372683      129:             10000001 ~ {1,8}
%e A372683      257:            100000001 ~ {1,9}
%e A372683      514:           1000000010 ~ {2,10}
%e A372683     1027:          10000000011 ~ {1,2,11}
%e A372683     2049:         100000000001 ~ {1,12}
%e A372683     4097:        1000000000001 ~ {1,13}
%e A372683     8193:       10000000000001 ~ {1,14}
%e A372683    16385:      100000000000001 ~ {1,15}
%e A372683    32770:     1000000000000010 ~ {2,16}
%e A372683    65537:    10000000000000001 ~ {1,17}
%e A372683   131073:   100000000000000001 ~ {1,18}
%e A372683   262145:  1000000000000000001 ~ {1,19}
%e A372683   524289: 10000000000000000001 ~ {1,20}
%t A372683 Table[NestWhile[#+1&,2^n,!SquareFreeQ[#]&],{n,0,10}]
%o A372683 (PARI) a(n) = my(k=2^n); while (!issquarefree(k), k++); k; \\ _Michel Marcus_, May 29 2024
%o A372683 (Python)
%o A372683 from itertools import count
%o A372683 from sympy import factorint
%o A372683 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 A372683 For primes instead of powers of two we have A112926, opposite A112925, sum A373197, length A373198.
%Y A372683 Counting zeros instead of all bits gives A372473, firsts of A372472.
%Y A372683 These are squarefree numbers at indices A372540, firsts of A372475.
%Y A372683 Counting ones instead of all bits gives A372541, firsts of A372433.
%Y A372683 The opposite (greatest squarefree number <= 2^n) is A372889.
%Y A372683 The difference from 2^n is A373125.
%Y A372683 For prime instead of squarefree we have:
%Y A372683 - bits A372684, firsts of A035100
%Y A372683 - zeros A372474, firsts of A035103
%Y A372683 - ones A372517, firsts of A014499
%Y A372683 A000120 counts ones in binary expansion (binary weight), zeros A080791.
%Y A372683 A005117 lists squarefree numbers.
%Y A372683 A030190 gives binary expansion, reversed A030308, length A070939 or A029837.
%Y A372683 A061398 counts squarefree numbers between primes (exclusive).
%Y A372683 A077643 counts squarefree terms between powers of 2, run-lengths of A372475.
%Y A372683 A143658 counts squarefree numbers up to 2^n.
%Y A372683 Cf. A029931, A048793, A049095, A049096, A059015, A069010, A076259, A077641, A211997, A230877.
%K A372683 nonn
%O A372683 0,2
%A A372683 _Gus Wiseman_, May 26 2024