A354758 -id:A354758 - cp's OEIS frontend

A354757 a(n) = Sum_{k = ceiling(n/2)..n-1} A354169(k).

0, 0, 1, 2, 6, 12, 15, 27, 59, 115, 127, 252, 508, 1004, 1021, 2013, 2047, 4031, 8127, 16307, 16375, 32631, 32767, 65279, 130815, 261375, 262143, 524270, 1048558, 2096110, 2097135, 4194253, 4194271, 8386527, 8388607, 16773119, 33550335, 67096575, 67108863

Offset: 0

Rémy Sigrist, Jun 06 2022

The 1's in the binary expansion of a(n) are forbidden in that of A354169(n). In other words, a(n) AND A354169(n) = 0 (where AND denotes the bitwise AND operator).

			a(5) = A354169(3) + A354169(4) = 4 + 8 = 12.
a(7) = A354169(4) + A354169(5) + A354169(6) = 8 + 3 + 16 = 27.

N. J. A. Sloane, Table of n, a(n) for n = 0..4800
Rémy Sigrist, Binary plot of a(n) (blue pixels) and A354169(n) (red pixels) for n=0..1000
Rémy Sigrist, PARI program

Cf. A354169, A354758.

A354780 is a bisection.

PARI
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See Links section.
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from itertools import count, islice
from collections import deque
from functools import reduce
from operator import or_
def A354757_gen(): # generator of terms
    aset, aqueue, b, f = {0,1,2}, deque([2]), 2, False
    yield from (0,0,1)
    while True:
        for k in count(1):
            m, j, j2, r, s = 0, 0, 1, b, k
            while r > 0:
                r, q = divmod(r,2)
                if not q:
                    s, y = divmod(s,2)
                    m += y*j2
                j += 1
                j2 *= 2
            if s > 0:
                m += s*2**b.bit_length()
            if m not in aset:
                yield sum(aqueue)
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = reduce(or_,aqueue)
                f = not f
                break
A354757_list = list(islice(A354757_gen(),40)) # Chai Wah Wu, Jun 06 2022

A355057 a(n) = product of prohibited prime factors of A090252(n).

Original entry on oeis.org

1, 1, 2, 6, 15, 30, 70, 210, 462, 6006, 51051, 102102, 277134, 6374082, 10623470, 223092870, 588153930, 18232771830, 51893273670, 2127624220470, 5381637734130, 252936973504110, 6702829797858915, 13405659595717830, 41628100849860630, 2539314151841498430, 7397132529277408470

Offset: 1

Michael De Vlieger, Jun 16 2022

nonn

Let s(n) = A090252(n) and let K(n) = A007947(n) = squarefree kernel of n.
Prime p | s(n) implies p does not divide s(n+j), 1 <= j <= n.
Therefore a(n) is the product of primes p that cannot divide s(n).
a(n) = product of distinct primes that divide a(j) for floor((n+1)/2) <= j <= n-1. - N. J. A. Sloane, Jun 17 2022

			a(1) = 1;
a(2) = 1 since s(1) = 1, and (2-1)/2 is not an integer;
a(3) = a(2) * K(s(2)) / K(s((3-1)/2)) = 1 * 2 / 1 = 2;
a(4) = a(3) * K(s(3)) = 2 * 3 = 6;
a(5) = a(4) * K(s(4)) / K(s((5-1)/2)) = 6 * 5 / 2 = 15;
a(6) = a(5) * K(s(5)) = 15 * 2 = 30;
a(7) = a(6) * K(s(6)) / K(s((7-1)/2)) = 30 * 7 / 3 = 70;
etc.

N. J. A. Sloane, Table of n, a(n) for n = 1..500
Michael De Vlieger, Annotated chart of prime factors of a(n), n = 1..64, plotting prime p | a(n) at (n, pi(p)) in black, and prime q | s(n) at (n, pi(q)) in red.
Michael De Vlieger, Plot of prime p | a(n), n = 1..2^12, plotting p | a(n) at (n, pi(p)) in black.

See A354758 for another version.
A354765 is a binary encoding.
Cf. A007947, A090252, A354764.

# To get first M terms, from N. J. A. Sloane, Jun 18 2022
with(numtheory);
M:=20; ans:=[1,1,2];
for i from 4 to M do
S:={}; j1:=floor((i+1)/2); j2:=i-1;
  for j from j1 to j2 do S:={op(S), op(factorset(b252[j]))} od:
t2 := product(S[k], k = 1..nops(S));
ans:=[op(ans),t2];
od:
ans;

Block[{s = Import["https://oeis.org/A090252/b090252.txt", "Data"][[1 ;; 120, -1]], m = 1}, Reap[Do[m *= Times @@ FactorInteger[s[[If[# == 0, 1, #] &[i - 1]]]][[All, 1]]; If[IntegerQ[#] && # > 0, m /= Times @@ FactorInteger[s[[#]]][[All, 1]]] &[(i - 1)/2]; Sow[m], {i, Length[s]}] ][[-1, -1]] ]

from math import prod, lcm, gcd
from itertools import count, islice
from collections import deque
from sympy import primefactors
def A355057_gen(): # generator of terms
    aset, aqueue, c, b, f = {1}, deque([1]), 2, 1, True
    yield 1
    while True:
        for m in count(c):
            if m not in aset and gcd(m,b) == 1:
                yield prod(primefactors(b))
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = lcm(*aqueue)
                f = not f
                while c in aset:
                    c += 1
                break
A355057_list = list(islice(A355057_gen(),20)) # Chai Wah Wu, Jun 18 2022

a(n) = a(n-1) * K(s(n-1)) / K(s((n-1)/2)), where the last operation is only carried out iff (n-1)/2 is an integer.

A354780 a(n) is the bitwise OR of (the binary expansions of) b(n+1) to b(2*n), where b is A354169.

Original entry on oeis.org

2, 12, 27, 115, 252, 1004, 2013, 4031, 16307, 32631, 65279, 261375, 524270, 2096110, 4194253, 8386527, 16773119, 67096575, 134217659, 536854459, 1073741623, 2147450751, 4294901759, 17179672575, 34359737599, 137438690559, 274877382143, 549754765311, 2199022205950, 4398044412927, 8796093022189, 35184367894509, 70368744175567

Offset: 1

N. J. A. Sloane, Jul 05 2022

If the binary expansion of a(n) has a 1 in the 2^i's bit (for any i >= 0) then A354169(2*n+1) must have a 0 in that bit.
A354169(2*n+1) is the smallest number not yet in A354169 which satisfies that condition (this follows at once from the definition of A354169).
This sequence bears the same relation to A354169 as A355057 does to A090252.

			Consider n=6. Then b(7) to b(12) are 32, 64, 12, 128, 256, 512. The bitwise OR of those 6 numbers is 1111101100_2 = 1004_10 = a(6). The bitwise complement of 1004_10 is 10011_2 = 19_10 = A354781(6), and A354169(6) = 17_10 = 10001_2.
On the other hand, for n=5, b(6) to b(10) are 16, 32, 64, 12, 128, whose bitwise OR is 11111100_2 = 252_10 = a(5). The bitwise complement of 252_10 is 3_10 = 11_2 = A354781(5). However, 3 has already appeared in A354169, and the smallest available number whose binary expansion is disjoint from 252_10 = 11111100_2 is 2^8 = 100000000_2 = 256_10 = 2^8 = A354169(5).

N. J. A. Sloane, Table of n, a(n) for n = 1..256

A bisection of A354757.
Cf. A354169, A354680, A354757, A354758, A354767, A354773, A354774, A354778, A355150, A354781.
See also A090252, A355057.

A354781 If the binary expansion of A354780(n) is 1 d_1 d_2 ... d_k, then the binary expansion of a(n) is c_1 c_2 ... c_k, where c_i = 1 - d_i.

Original entry on oeis.org

1, 3, 4, 12, 3, 19, 34, 64, 76, 136, 256, 768, 17, 1041, 50, 2080, 4096, 12288, 68, 16452, 200, 32896, 65536, 196608, 768, 262912, 524800, 1048576, 1049601, 2098176, 18, 4194322, 2096, 8390656, 16777216, 50331648, 12288, 67121152, 134225920, 268435456, 268451844, 536887296, 72, 1073741896, 32960, 2147516416, 4294967296, 12884901888

Offset: 1

N. J. A. Sloane, Jul 05 2022

			See A354780.

N. J. A. Sloane, Table of n, a(n) for n = 1..2400

Cf. A354169, A354680, A354757, A354758, A354767, A354773, A354774, A354778, A355150, A354780.

cp's OEIS Frontend

A354757 a(n) = Sum_{k = ceiling(n/2)..n-1} A354169(k).

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A355057 a(n) = product of prohibited prime factors of A090252(n).

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A354780 a(n) is the bitwise OR of (the binary expansions of) b(n+1) to b(2*n), where b is A354169.

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A354781 If the binary expansion of A354780(n) is 1 d_1 d_2 ... d_k, then the binary expansion of a(n) is c_1 c_2 ... c_k, where c_i = 1 - d_i.

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