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A360573 Odd numbers with exactly three zeros in their binary expansion.

%I A360573 #56 Jun 16 2025 23:49:08
%S A360573 17,35,37,41,49,71,75,77,83,85,89,99,101,105,113,143,151,155,157,167,
%T A360573 171,173,179,181,185,199,203,205,211,213,217,227,229,233,241,287,303,
%U A360573 311,315,317,335,343,347,349,359,363,365,371,373,377,399,407,411,413
%N A360573 Odd numbers with exactly three zeros in their binary expansion.
%C A360573 If m is a term then 2*m+1 is another term, since if M is the binary expansion of m, then M.1 where . stands for concatenation is the binary expansion of 2*m+1.
%C A360573 A052996 \ {1,3,8} is a subsequence, since for m >= 3, A052996(m) = 9*2^(m-2) - 1 has 100011..11 with m-2 trailing 1 for binary expansion.
%C A360573 A171389 \ {20} is a subsequence, since for m >= 1, A171389(m) = 21*2^m - 1 has 1010011..11 with m trailing 1 for binary expansion.
%C A360573 A198276 \ {18} is a subsequence, since for m >= 1, A198276(m) = 19*2^m - 1 has 1001011..11 with m trailing 1 for binary expansion.
%C A360573 Binary expansion of a(n) is A360574(n).
%C A360573 {8*a(n), n>0} form a subsequence of A353654 (numbers with three trailing 0 bits and three other 0 bits).
%C A360573 Numbers of the form 2^(a+1) - 2^b - 2^c - 2^d - 1 where a > b > c > d > 0. - _Robert Israel_, Feb 13 2023
%F A360573 A023416(a(n)) = 3.
%F A360573 Let a = floor((24n)^(1/4))+4 if n>binomial(floor((24n)^(1/4))+2,4) and a = floor((24n)^(1/4))+3 otherwise. Let j = binomial(a-1,4)-n. Then a(n) = 2^a-1-2^(A360010(j+1)+2)-2^(A056557(j)+2)-2^(A333516(j+1)). - _Chai Wah Wu_, Dec 18 2024
%e A360573 35_10 = 100011_2, so 35 is a term.
%p A360573 q:= n-> n::odd and add(1-i, i=Bits[Split](n))=3:
%p A360573 select(q, [$1..575])[];  # _Alois P. Heinz_, Feb 12 2023
%p A360573 # Alternative:
%p A360573 [seq(seq(seq(seq(2^(a+1) - 2^b - 2^c - 2^d - 1, d = c-1..1,-1), c=b-1..2,-1),b=a-1..3,-1),a=4..12)]; # _Robert Israel_, Feb 13 2023
%t A360573 Select[Range[1, 500, 2], DigitCount[#, 2, 0] == 3 &] (* _Amiram Eldar_, Feb 12 2023 *)
%o A360573 (Python)
%o A360573 def ok(n): return n&1 and bin(n)[2:].count("0") == 3
%o A360573 print([k for k in range(414) if ok(k)]) # _Michael S. Branicky_, Feb 12 2023
%o A360573 (Python)
%o A360573 from itertools import count, islice
%o A360573 from sympy.utilities.iterables import multiset_permutations
%o A360573 def A360573_gen(): # generator of terms
%o A360573     yield from (int('1'+''.join(d)+'1',2) for l in count(0) for d in  multiset_permutations('000'+'1'*l))
%o A360573 A360573_list = list(islice(A360573_gen(),54)) # _Chai Wah Wu_, Feb 18 2023
%o A360573 (Python)
%o A360573 from itertools import combinations, count, islice
%o A360573 def agen(): yield from ((1<<m)-(1<<i)-(1<<j)-(1<<k)-1 for m in count(5) for i, j, k in combinations(range(m-2,0,-1), 3))
%o A360573 print(list(islice(agen(), 54))) # _Michael S. Branicky_, Feb 18 2023
%o A360573 (Python)
%o A360573 from math import comb, isqrt
%o A360573 from sympy import integer_nthroot
%o A360573 def A056557(n): return (k:=isqrt(r:=n+1-comb((m:=integer_nthroot(6*(n+1), 3)[0])-(n<comb(m+2, 3))+2, 3)<<1))-((r<<2)<=(k<<2)*(k+1)+1)
%o A360573 def A333516(n): return (r:=n-1-comb((m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))+1, 3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)), 2)+1
%o A360573 def A360010(n): return (m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))
%o A360573 def A360573(n):
%o A360573     a = (a2:=integer_nthroot(24*n, 4)[0])+(n>comb(a2+2, 4))+3
%o A360573     j = comb(a-1,4)-n
%o A360573     b, c, d = A360010(j+1)+2, A056557(j)+2, A333516(j+1)
%o A360573     return (1<<a)-(1<<b)-(1<<c)-(1<<d)-1 # _Chai Wah Wu_, Dec 18 2024
%o A360573 (PARI) isok(m) = (m%2) && #select(x->(x==0), binary(m)) == 3; \\ _Michel Marcus_, Feb 13 2023
%Y A360573 Cf. A005408, A023416, A056557, A333516, A353654, A360010, A360574.
%Y A360573 Subsequences: A052996 \ {1,3,8}, A171389 \ {20}, A198276 \ {18}.
%Y A360573 Odd numbers with k zeros in their binary expansion: A000225 (k=0), A190620 (k=1), A357773 (k=2), this sequence (k=3).
%K A360573 nonn,base
%O A360573 1,1
%A A360573 _Bernard Schott_, Feb 12 2023