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A372541 Least k such that the k-th squarefree number has exactly n ones in its binary expansion.

%I A372541 #24 May 11 2024 13:00:09
%S A372541 1,3,6,11,20,60,78,157,314,624,1245,3736,4982,9962,19920,39844,79688,
%T A372541 239046,318725,956194,1912371,2549834,5099650,15298984,20398664,
%U A372541 40797327,81594626,163189197,326378284,979135127,1305513583,2611027094,5222054081,10444108051
%N A372541 Least k such that the k-th squarefree number has exactly n ones in its binary expansion.
%H A372541 Chai Wah Wu, <a href="/A372541/b372541.txt">Table of n, a(n) for n = 0..56</a>
%H A372541 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hamming_weight">Hamming weight</a>.
%e A372541 The squarefree numbers A005117(a(n)) together with their binary expansions and binary indices begin:
%e A372541        1:                   1 ~ {1}
%e A372541        3:                  11 ~ {1,2}
%e A372541        7:                 111 ~ {1,2,3}
%e A372541       15:                1111 ~ {1,2,3,4}
%e A372541       31:               11111 ~ {1,2,3,4,5}
%e A372541       95:             1011111 ~ {1,2,3,4,5,7}
%e A372541      127:             1111111 ~ {1,2,3,4,5,6,7}
%e A372541      255:            11111111 ~ {1,2,3,4,5,6,7,8}
%e A372541      511:           111111111 ~ {1,2,3,4,5,6,7,8,9}
%e A372541     1023:          1111111111 ~ {1,2,3,4,5,6,7,8,9,10}
%e A372541     2047:         11111111111 ~ {1,2,3,4,5,6,7,8,9,10,11}
%e A372541     6143:       1011111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,13}
%e A372541     8191:       1111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13}
%e A372541    16383:      11111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14}
%e A372541    32767:     111111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
%e A372541    65535:    1111111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}
%e A372541   131071:   11111111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17}
%t A372541 nn=10000;
%t A372541 spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
%t A372541 dcs=DigitCount[Select[Range[nn],SquareFreeQ],2,1];
%t A372541 Table[Position[dcs,i][[1,1]],{i,spnm[dcs-1]}]
%o A372541 (Python)
%o A372541 from math import isqrt
%o A372541 from itertools import count
%o A372541 from sympy import factorint, mobius
%o A372541 from sympy.utilities.iterables import multiset_permutations
%o A372541 def A372541(n):
%o A372541     if n==0: return 1
%o A372541     for l in count(n):
%o A372541         m = 1<<l
%o A372541         for d in multiset_permutations('0'*(l-n)+'1'*n):
%o A372541             k = m+int('0'+''.join(d),2)
%o A372541             if max(factorint(k).values(),default=0)==1:
%o A372541                 return sum(mobius(a)*(k//a**2) for a in range(1, isqrt(k)+1)) # _Chai Wah Wu_, May 10 2024
%Y A372541 Positions of firsts appearances in A372433.
%Y A372541 Counting zeros instead of ones gives A372473, firsts in A372472.
%Y A372541 For prime instead of squarefree we have A372517, firsts of A014499.
%Y A372541 Counting bits (length) gives A372540, firsts of A372475, runs A077643.
%Y A372541 A000120 counts ones in binary expansion (binary weight), zeros A080791.
%Y A372541 A005117 lists squarefree numbers.
%Y A372541 A030190 gives binary expansion, reversed A030308.
%Y A372541 A048793 lists positions of ones in reversed binary expansion, sum A029931.
%Y A372541 A145037, A097110 count ones minus zeros, for primes A372516, A177796.
%Y A372541 A371571 lists positions of zeros in binary expansion, sum A359359.
%Y A372541 A371572 lists positions of ones in binary expansion, sum A230877.
%Y A372541 A372515 lists positions of zeros in reversed binary expansion, sum A359400.
%Y A372541 Cf. A023416, A049093, A049094, A069010, A070939, A071403, A211997, A280296, A372474.
%K A372541 nonn,base
%O A372541 0,2
%A A372541 _Gus Wiseman_, May 09 2024
%E A372541 a(23)-a(33) from _Chai Wah Wu_, May 10 2024