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

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 32, 34, 36, 40, 44, 48, 52, 56, 64, 66, 68, 72, 76, 80, 84, 88, 96, 100, 104, 112, 120, 128, 130, 132, 136, 140, 144, 148, 152, 160, 164, 168, 176, 184, 192, 196, 200, 208, 216, 224, 232, 240, 256, 258, 260, 264, 268

Offset

1,2

Comments

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

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

From David A. Corneth, May 31 2024: (Start)

If k is in the sequence then so is 2*k.

a(A000045(k)) = 2^(k-2) for k >= 2. (End)

Apart from a(1), all terms are even. - Paolo Xausa, May 31 2024

Links

Michel Marcus, Table of n, a(n) for n = 1..10945

Alistair Bird, Jozef Schreier, Schreier sets and the Fibonacci sequence, Out Of The Norm blog, May 13 2012.

Formula

a(n) = b(n)*A001316(b(n))/2 where b(n) = A048679(n).

a(n) = Sum_{i=0..n-1} 2^A213911(i).

a(n) = 2^(A072649(n) - 1) + [c(n) > 0]*2*a(c(n)) where c(n) = A066628(n).

a(n) = 2*a(A005206(n)) + [A003849(n) = 1]*2^A007895(n-1) for n > 1 with a(1) = 1.

Mathematica

Join[{1}, Select[Range[2, 1000, 2], DigitCount[#, 2, 1] <= IntegerExponent[#, 2] + 1 &]] (* Paolo Xausa, May 31 2024 *)

Prog

(PARI) isok(n) = hammingweight(n) <= (valuation(n, 2) + 1)
(PARI) b1(n) = if(n == 0, 0, my(A = 0); forstep(i = logint(n, 2), 0, -1, if(A >= i, break); if(bittest(n, i), A++)); (n >> A)*(1 << A))
upto(n, m) = my(A = b1(m), B, v1); v1 = vector(n, i, 0); for(i = 1, n, B = hammingweight(A) - 1; A += 2^(B - (A/2^B)%2 + 1); v1[i] = A); v1 \\ first n terms greater than m \\  [verification needed]
(PARI) Zeckendorf(n) = my(A = n, B1, B2, C, m = 1, v1); v1 = [n > 0]; while(fibonacci(m+1) <= n, m++); m--; B1 = fibonacci(m+1); B2 = fibonacci(m); A -= B1; while(m > 1, m--; C = 0; if(A >= B2, A -= B2; C = 1); [B1, B2] = [B2, B1 - B2]; v1 = concat(v1, C)); v1
a(n) = my(v1); v1 = Zeckendorf(n); for(i = 2, #v1, if(v1[i], v1 = concat(vector(#v1-1, j, v1[j + (j >= (i-1))]), 0))); fromdigits(v1, 2) \\  [verification needed]
(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);
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
(Python)
def ok(n): return n.bit_count() <= (-n&n).bit_length()
print([k for k in range(1, 300) if ok(k)]) # Michael S. Branicky, May 31 2024
(Python)  # Assuming the list starts with 0.
def a():
    n = na = nb = 1
    while True:
        yield not(nb < (na - 1) << 1)
        nb, na = na, n.bit_count()
        n += 1
aList = a(); print([n for n in range(77) if next(aList)]) # Peter Luschny, Jun 07 2024

Crossrefs

Cf. A000045, A000120, A001316, A001511, A003849, A005206, A007895, A048679, A066628, A072649, A213911.

Cf. A355489, A373345, A373347 (complement), A373557.

Sequence in context: A051178 A093891 A213708 * A239063 A151999 A267817

Adjacent sequences: A371173 A371174 A371175 * A371177 A371178 A371179

Keyword

nonn,base

Author

Mikhail Kurkov, Mar 14 2024

Status

approved