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

Author

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

Keywords

• nonn
• base

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}.

Examples

			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.
		

Links

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

Crossrefs

Cf. A090252, A355057.

Programs

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;

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