cp's OEIS Frontend

A372473 Least k such that the k-th squarefree number has exactly n zeros in its binary expansion.

%I A372473 #20 May 11 2024 12:59:55
%S A372473 1,2,7,12,21,40,79,158,315,1247,1246,2492,4983,9963,19921,39845,79689,
%T A372473 159361,318726,637462,1274919,2549835,5099651,10199302,20398665,
%U A372473 40797328,81594627,163189198,326378285,652756723,1305513584,2611027095,5222054082,10444108052
%N A372473 Least k such that the k-th squarefree number has exactly n zeros in its binary expansion.
%C A372473 Note that the data is not strictly increasing.
%H A372473 Chai Wah Wu, <a href="/A372473/b372473.txt">Table of n, a(n) for n = 0..57</a>
%e A372473 The squarefree numbers A005117(a(n)) together with their binary expansions and binary indices begin:
%e A372473      1:              1 ~ {1}
%e A372473      2:             10 ~ {2}
%e A372473     10:           1010 ~ {2,4}
%e A372473     17:          10001 ~ {1,5}
%e A372473     33:         100001 ~ {1,6}
%e A372473     65:        1000001 ~ {1,7}
%e A372473    129:       10000001 ~ {1,8}
%e A372473    257:      100000001 ~ {1,9}
%e A372473    514:     1000000010 ~ {2,10}
%e A372473   2051:   100000000011 ~ {1,2,12}
%e A372473   2049:   100000000001 ~ {1,12}
%e A372473   4097:  1000000000001 ~ {1,13}
%e A372473   8193: 10000000000001 ~ {1,14}
%t A372473 nn=10000;
%t A372473 spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
%t A372473 dcs=DigitCount[Select[Range[nn],SquareFreeQ],2,0];
%t A372473 Table[Position[dcs,i][[1,1]],{i,0,spnm[dcs]}]
%o A372473 (Python)
%o A372473 from math import isqrt
%o A372473 from itertools import count
%o A372473 from sympy import factorint, mobius
%o A372473 from sympy.utilities.iterables import multiset_permutations
%o A372473 def A372473(n):
%o A372473     if n==0: return 1
%o A372473     for l in count(n):
%o A372473         m = 1<<l
%o A372473         for d in multiset_permutations('0'*n+'1'*(l-n)):
%o A372473             k = m+int('0'+''.join(d),2)
%o A372473             if max(factorint(k).values(),default=0)==1:
%o A372473                  return sum(mobius(a)*(k//a**2) for a in range(1, isqrt(k)+1)) # _Chai Wah Wu_, May 10 2024
%Y A372473 Positions of first appearances in A372472.
%Y A372473 For prime instead of squarefree we have A372474, A035103, A372517, A014499.
%Y A372473 Counting bits (length) gives A372540, firsts of A372475, runs A077643.
%Y A372473 Counting 1's (weight) instead of 0's gives A372541, firsts of A372433.
%Y A372473 A000120 counts ones in binary expansion (binary weight), zeros A080791.
%Y A372473 A005117 lists squarefree numbers.
%Y A372473 A030190 gives binary expansion, reversed A030308.
%Y A372473 A048793 lists positions of ones in reversed binary expansion, sum A029931.
%Y A372473 A070939 gives length of binary expansion (number of bits).
%Y A372473 A371571 lists positions of zeros in binary expansion, sum A359359.
%Y A372473 A371572 lists positions of ones in binary expansion, sum A230877.
%Y A372473 A372515 lists positions of zeros in reversed binary expansion, sum A359400.
%Y A372473 Cf. A035100, A039004, A049093, A049094, A059015, A069010, A145037, A211997.
%K A372473 nonn,base
%O A372473 0,2
%A A372473 _Gus Wiseman_, May 09 2024
%E A372473 a(23)-a(33) from _Chai Wah Wu_, May 10 2024