|
%I #25 Aug 26 2024 20:07:33
%S 1,2,4,7,12,21,40,79,158,315,625,1246,2492,4983,9963,19921,39845,
%T 79689,159361,318726,637462,1274919,2549835,5099651,10199302,20398665,
%U 40797328,81594627,163189198,326378285,652756723,1305513584,2611027095,5222054082,10444108052
%N Least k such that the k-th squarefree number has binary expansion of length n. Index of the smallest squarefree number >= 2^n.
%H Chai Wah Wu, <a href="/A372540/b372540.txt">Table of n, a(n) for n = 0..73</a>
%F A005117(a(n)) = A372683(n).
%F a(n) = A143658(n)+1 for n > 1. - _Chai Wah Wu_, Aug 26 2024
%e The squarefree numbers A005117(a(n)) 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}
%t nn=1000;
%t ssnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[Max@@#]&];
%t dcs=IntegerLength[Select[Range[nn],SquareFreeQ],2];
%t Table[Position[dcs,i][[1,1]],{i,ssnm[dcs]}]
%o (Python)
%o from itertools import count
%o from math import isqrt
%o from sympy import mobius, factorint
%o def A372540(n): return next(sum(mobius(a)*(k//a**2) for a in range(1, isqrt(k)+1)) for k in count(1<<n) if max(factorint(k).values(),default=0)==1) if n else 1 # _Chai Wah Wu_, May 12 2024
%Y Counting zeros instead of length gives A372473, firsts of A372472.
%Y For prime instead of squarefree we have:
%Y - zeros A372474, firsts of A035103
%Y - ones A372517, firsts of A014499
%Y - bits A372684, firsts of A035100
%Y Positions of first appearances in A372475, run-lengths A077643.
%Y For weight instead of length we have A372541, firsts of A372433.
%Y Indices of the squarefree numbers listed by A372683.
%Y A000120 counts ones in binary expansion (binary weight), zeros A080791.
%Y A005117 lists squarefree numbers.
%Y A030190 gives binary expansion, reversed A030308.
%Y A070939 counts bits, binary length, or length of binary expansion.
%Y Cf. A029931, A048793, A049093, A049094, A059015, A069010, A143658, A211997, A230877.
%K nonn,base
%O 0,2
%A _Gus Wiseman_, May 10 2024
%E a(24)-a(34) from _Chai Wah Wu_, May 12 2024
| 