cp's OEIS Frontend

A371176 Numbers k such that A000120(k) <= A001511(k).

%I A371176 #95 Aug 13 2025 03:12:22
%S A371176 1,2,4,6,8,10,12,16,18,20,24,28,32,34,36,40,44,48,52,56,64,66,68,72,
%T A371176 76,80,84,88,96,100,104,112,120,128,130,132,136,140,144,148,152,160,
%U A371176 164,168,176,184,192,196,200,208,216,224,232,240,256,258,260,264,268
%N A371176 Numbers k such that A000120(k) <= A001511(k).
%C A371176 It appears that this sequence is obtained when ordering Schreier sets as explained in the Bird link. See decM(n) PARI code. - _Michel Marcus_, May 31 2024
%C A371176 That is correct since the binary representation of these numbers can be put into 1-to-1 correspondence with Schreier sets, which satisfy |X| <= min X, using the indicator function of X as the bits (starting from the right, LSB). The reason is that A000120 then computes |X| and A001511 computes min X. For example, the Schreier set X = {2, 5} can be mapped to 10010_2 = 18. - _Michael S. Branicky_, May 31 2024
%C A371176 From _David A. Corneth_, May 31 2024: (Start)
%C A371176 If k is in the sequence then so is 2*k.
%C A371176 a(A000045(k)) = 2^(k-2) for k >= 2. (End)
%C A371176 Apart from a(1), all terms are even. - _Paolo Xausa_, May 31 2024
%C A371176 Zeckendorf representation of n with rewrite 0 -> 0, {0, 1} -> 1 and k-1 zeros appended to the right side (where k is the number of ones in the given representation) and then interpreted as binary expansion is the same as a(n) (see the first formula). - _Mikhail Kurkov_, Oct 21 2024
%H A371176 Michel Marcus, <a href="/A371176/b371176.txt">Table of n, a(n) for n = 1..10945</a>
%H A371176 Alistair Bird, <a href="https://outofthenormmaths.wordpress.com/2012/05/13/jozef-schreier-schreier-sets-and-the-fibonacci-sequence/">Jozef Schreier, Schreier sets and the Fibonacci sequence</a>, Out Of The Norm blog, May 13 2012.
%H A371176 Vaclav Kotesovec, <a href="/A371176/a371176_1.jpg">Plot of log(a(n))/log(n) for n = 2..2129243</a>
%H A371176 Vaclav Kotesovec, <a href="/A371176/a371176_2.jpg">Plot of a(n) / 2^(log(sqrt(5)*n)/log((1 + sqrt(5))/2) - 2) for n = 2..2129243</a>
%F A371176 a(n) = b(n)*A001316(b(n))/2 where b(n) = A048679(n).
%F A371176 a(n) = Sum_{i=0..n-1} 2^A213911(i).
%F A371176 a(n) = 2^(A072649(n) - 1) + [c(n) > 0]*2*a(c(n)) where c(n) = A066628(n).
%F A371176 a(n) = 2*a(A005206(n)) - A003849(n)*2^A007895(n-1) for n > 1 with a(1) = 1.
%F A371176 Conjecture: lim sup_{n -> oo} log(a(n))/log(n) = log(2) / log((1 + sqrt(5))/2) = 1.440420090412556479... = A104287. - _Vaclav Kotesovec_, Aug 12 2025
%p A371176 filter:= proc(n) convert(convert(n,base,2),`+`) <= 1+padic:-ordp(n,2) end proc:
%p A371176 select(filter, [1,seq(i,i=2..1000,2)]); # _Robert Israel_, Oct 20 2024
%t A371176 Join[{1}, Select[Range[2, 1000, 2], DigitSum[#, 2] <= IntegerExponent[#, 2] + 1 &]] (* _Paolo Xausa_, Aug 12 2025 *)
%o A371176 (PARI) isok(n) = hammingweight(n) <= (valuation(n, 2) + 1)
%o A371176 (PARI) M(n) = my(list=List()); for (i=1, n, forsubset(i, s, my(bOk = if (#s && (vecmax(s) == n), #s <= vecmin(s), 0)); if (bOk, listput(list, vecsort(Vec(s),,4))););); Vec(list);
%o A371176 decM(nn) = my(v = vector(nn, k, M(k)), list=List()); for (i=1, #v, my(vi = v[i]); for (j=1, #vi, my(s = vecsort(vi[j]), slist=List(), m = vecmax(s)); forstep(k=m, 1, -1, listput(slist, sign(vecsearch(s, k)))); listput(list, fromdigits(Vec(slist), 2)););); vecsort(Vec(list)); \\ _Michel Marcus_, May 31 2024
%o A371176 (Python)
%o A371176 def ok(n): return n.bit_count() <= (-n&n).bit_length()
%o A371176 print([k for k in range(1, 300) if ok(k)]) # _Michael S. Branicky_, May 31 2024
%o A371176 (Python)  # Assuming the list starts with 0.
%o A371176 def a():
%o A371176     n = na = nb = 1
%o A371176     while True:
%o A371176         yield not(nb < (na - 1) << 1)
%o A371176         nb, na = na, n.bit_count()
%o A371176         n += 1
%o A371176 aList = a(); print([n for n in range(77) if next(aList)]) # _Peter Luschny_, Jun 07 2024
%Y A371176 Cf. A000045, A000120, A001316, A001511, A003849, A005206, A007895, A048679, A066628, A072649, A213911.
%Y A371176 Cf. A355489, A373345, A373347 (complement), A373557.
%Y A371176 Cf. A104287.
%K A371176 nonn,base
%O A371176 1,2
%A A371176 _Mikhail Kurkov_, Mar 14 2024