A354765 a(n) is a binary encoded version of A355057(n).

0, 0, 1, 3, 6, 7, 13, 15, 27, 59, 122, 123, 243, 499, 501, 511, 1007, 2031, 4047, 8143, 16271, 32655, 65422, 65423, 130831, 261903, 523791, 1048079, 2096651, 2096671, 4193813, 4193815, 4193311, 8387615, 16775199, 33552415, 67104799, 134213663, 268427295, 536862751, 1073725471, 2147467295, 4294934559, 8589901855, 17179803679

Offset

1,4

Comments

Let plist = list of forbidden primes for A090252(n); A355057(n) is the product of these primes. Then a(n) = Sum of 2^(i-1) over all prime(i) in plist.

Conversely, if a(n) has binary expansion a(n) = Sum b(i)*2^i, b(i) = 0 or 1, then plist consists of {prime(i+1) such that b(i) = 1}.

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..1000

Example

For n = 7 the forbidden primes are 2, 5, 7 = prime(1), prime(3) and prime(4). Their product is A355057(7) = 70. Then a(7) = 2^0 + 2^2 + 2^3 = 13.

Maple

# To get first M terms:
with(numtheory);
M:=20; ans:=[0, 0, 1];
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:
plis:=sort(convert(S, list));
t3:=0; for ii from 1 to nops(plis) do p:=plis[ii]; p2:=pi(p); t3:=t3+2^(p2-1); od:
ans:=[op(ans), t3];
od:
ans;

Prog

(Python)
from math import gcd, lcm
from itertools import count, islice
from collections import deque
from sympy import primepi, primefactors
def A354765_gen(): # generator of terms
    aset, aqueue, c, b, f = {1}, deque([1]), 2, 1, True
    yield 0
    while True:
        for m in count(c):
            if m not in aset and gcd(m, b) == 1:
                yield sum(2**(primepi(p)-1) for p in 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
A354765_list = list(islice(A354765_gen(), 20)) # Chai Wah Wu, Jun 18 2022

Crossrefs

Cf. A090252, A355057.

Sequence in context: A137595 A033053 A248388 * A107850 A216514 A051218

Adjacent sequences: A354762 A354763 A354764 * A354766 A354767 A354768

Keyword

nonn,base

Author

Michael De Vlieger and N. J. A. Sloane, Jun 18 2022

Status

approved