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a(n) is a binary encoded version of A355057(n).
4

%I #30 Jun 19 2022 06:23:42

%S 0,0,1,3,6,7,13,15,27,59,122,123,243,499,501,511,1007,2031,4047,8143,

%T 16271,32655,65422,65423,130831,261903,523791,1048079,2096651,2096671,

%U 4193813,4193815,4193311,8387615,16775199,33552415,67104799,134213663,268427295,536862751,1073725471,2147467295,4294934559,8589901855,17179803679

%N a(n) is a binary encoded version of A355057(n).

%C 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.

%C 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}.

%H N. J. A. Sloane, <a href="/A354765/b354765.txt">Table of n, a(n) for n = 1..1000</a>

%e 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.

%p # To get first M terms:

%p with(numtheory);

%p M:=20; ans:=[0,0,1];

%p for i from 4 to M do

%p S:={}; j1:=floor((i+1)/2); j2:=i-1;

%p for j from j1 to j2 do S:={op(S), op(factorset(b252[j]))} od:

%p plis:=sort(convert(S,list));

%p t3:=0; for ii from 1 to nops(plis) do p:=plis[ii]; p2:=pi(p); t3:=t3+2^(p2-1); od:

%p ans:=[op(ans),t3];

%p od:

%p ans;

%o (Python)

%o from math import gcd, lcm

%o from itertools import count, islice

%o from collections import deque

%o from sympy import primepi, primefactors

%o def A354765_gen(): # generator of terms

%o aset, aqueue, c, b, f = {1}, deque([1]), 2, 1, True

%o yield 0

%o while True:

%o for m in count(c):

%o if m not in aset and gcd(m,b) == 1:

%o yield sum(2**(primepi(p)-1) for p in primefactors(b))

%o aset.add(m)

%o aqueue.append(m)

%o if f: aqueue.popleft()

%o b = lcm(*aqueue)

%o f = not f

%o while c in aset:

%o c += 1

%o break

%o A354765_list = list(islice(A354765_gen(),20)) # _Chai Wah Wu_, Jun 18 2022

%Y Cf. A090252, A355057.

%K nonn,base

%O 1,4

%A _Michael De Vlieger_ and _N. J. A. Sloane_, Jun 18 2022