A348367 -id:A348367 - cp's OEIS frontend

A348368 Numbers k such that w(k + w(k)) < w(k), where w(k) is the binary weight of k, A000120(k).

6, 7, 13, 14, 15, 21, 29, 30, 31, 37, 45, 46, 47, 55, 59, 60, 61, 62, 63, 69, 77, 78, 79, 87, 91, 92, 93, 94, 95, 103, 107, 108, 109, 111, 115, 123, 124, 125, 126, 127, 133, 141, 142, 143, 151, 155, 156, 157, 158, 159, 167, 171, 172, 173, 175, 179, 187, 188, 189

Offset: 1

Ctibor O. Zizka, Oct 15 2021

			k = 91; A000120(91 + A000120(91)) < A000120(91), thus k = 91 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000120, A092391, A348367.

q:= n-> (wt-> is(wt(n+wt(n)) add(i, i=Bits[Split](k))):
select(q, [$0..200])[];  # Alois P. Heinz, Oct 15 2021

h[n_] := DigitCount[n, 2, 1]; q[n_] := h[n + (hn = h[n])] < hn; Select[Range[200], q] (* Amiram Eldar, Oct 15 2021 *)

def h(n): return bin(n).count('1')
def ok(n): return h(n + h(n)) < h(n)
print(list(filter(ok, range(1, 190)))) # Michael S. Branicky, Oct 15 2021

k : A000120(A092391(k)) < A000120(k); A348367(k) < A000120(k).

A352776 Numbers k such that w(k + w(k)) = w(k), where w(k) is the binary weight of k, A000120(k).

Original entry on oeis.org

0, 1, 3, 10, 11, 18, 19, 22, 23, 25, 34, 35, 38, 39, 41, 49, 53, 54, 66, 67, 70, 71, 73, 81, 85, 86, 97, 101, 102, 110, 116, 117, 119, 130, 131, 134, 135, 137, 145, 149, 150, 161, 165, 166, 174, 180, 181, 183, 193, 197, 198, 206, 212, 213, 215, 228, 229, 231, 236, 237, 243, 246, 247, 258, 259, 262, 263, 265, 273

Offset: 1

Ctibor O. Zizka, Apr 02 2022

w(k + w(k)) - w(k) = 0 this sequence, w(k + w(k)) - w(k) = 2 for k = 4*j, where A000120(j) = 3.

			k = 18; A000120(18 + A000120(18)) = A000120(18), thus k = 18 is a term.

Cf. A000120, A092391, A348367.

w[n_] := DigitCount[n, 2, 1]; Select[Range[0, 300], w[# + w[#]] == w[#] &] (* Amiram Eldar, Apr 02 2022 *)

def w(n): return bin(n).count("1")
def ok(n): wn = w(n); return w(n + wn) == wn
print([k for k in range(274) if ok(k)]) # Michael S. Branicky, Apr 02 2022

k : A000120(A092391(k)) = A000120(k); A348367(k) = A000120(k).

A352778 Numbers k such that w(k + w(k)) > w(k), where w(k) is the binary weight of k, A000120(k).

Original entry on oeis.org

2, 4, 5, 8, 9, 12, 16, 17, 20, 24, 26, 27, 28, 32, 33, 36, 40, 42, 43, 44, 48, 50, 51, 52, 56, 57, 58, 64, 65, 68, 72, 74, 75, 76, 80, 82, 83, 84, 88, 89, 90, 96, 98, 99, 100, 104, 105, 106, 112, 113, 114, 118, 120, 121, 122, 128, 129, 132, 136, 138, 139, 140, 144, 146, 147, 148, 152, 153, 154, 160, 162, 163, 164

Offset: 1

Ctibor O. Zizka, Apr 02 2022

			k = 17; A000120(17 + A000120(17)) > A000120(17), thus k = 17 is a term.

Cf. A000120, A092391, A348367.

w[n_] := DigitCount[n, 2, 1]; Select[Range[200], w[# + w[#]] > w[#] &] (* Amiram Eldar, Apr 02 2022 *)

def w(n): return bin(n).count("1")
def ok(n): wn = w(n); return w(n + wn) > wn
print([k for k in range(165) if ok(k)]) # Michael S. Branicky, Apr 02 2022

k : A000120(A092391(k)) > A000120(k); A348367(k) > A000120(k).

cp's OEIS Frontend

A348368 Numbers k such that w(k + w(k)) < w(k), where w(k) is the binary weight of k, A000120(k).

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A352776 Numbers k such that w(k + w(k)) = w(k), where w(k) is the binary weight of k, A000120(k).

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A352778 Numbers k such that w(k + w(k)) > w(k), where w(k) is the binary weight of k, A000120(k).

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Mathematica

Python

Formula