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A329873 a(n) is the number of distinct prime numbers whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.

%I A329873 #14 Jan 26 2022 11:55:06
%S A329873 0,0,1,1,1,3,2,2,1,3,3,5,2,5,3,2,1,4,3,6,3,6,5,6,2,5,5,6,3,6,3,3,1,4,
%T A329873 4,7,3,9,6,7,3,8,6,9,5,8,6,8,2,6,5,7,5,8,6,8,3,6,6,9,3,8,4,3,1,4,4,8,
%U A329873 4,9,7,9,3,11,9,11,6,11,7,10,3,8,8,12,6
%N A329873 a(n) is the number of distinct prime numbers whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.
%C A329873 This sequence is unbounded.
%H A329873 Rémy Sigrist, <a href="/A329873/b329873.txt">Table of n, a(n) for n = 0..16384</a>
%F A329873 A078826(n) <= a(n) <= A007306(n+1).
%F A329873 a(2*n) = a(n) + A036987(n) for any n > 0.
%F A329873 a(2^n) = 1 for any n > 0.
%e A329873 The first terms, alongside the binary representations of n and of the corresponding prime numbers, are:
%e A329873   n   a(n)  bin(n)  {bin(p)}
%e A329873   --  ----  ------  --------------------
%e A329873    0     0       0  {}
%e A329873    1     0       1  {}
%e A329873    2     1      10  {10}
%e A329873    3     1      11  {11}
%e A329873    4     1     100  {10}
%e A329873    5     3     101  {10, 11, 101}
%e A329873    6     2     110  {10, 11}
%e A329873    7     2     111  {11, 111}
%e A329873    8     1    1000  {10}
%e A329873    9     3    1001  {10, 11, 101}
%e A329873   10     3    1010  {10, 11, 101}
%e A329873   11     5    1011  {10, 11, 101, 111, 1011}
%e A329873   12     2    1100  {10, 11}
%p A329873 b:= proc(n) option remember; `if`(n=0, {0},
%p A329873       map(x-> [x, 2*x+r][], b(iquo(n, 2, 'r'))))
%p A329873     end:
%p A329873 a:= n-> nops(select(isprime, b(n))):
%p A329873 seq(a(n), n=0..84);  # _Alois P. Heinz_, Jan 26 2022
%o A329873 (PARI) a(n,base=2) = { my (b=digits(n,base), s=[0]); for (k=1, #b, s = setunion(s, apply(o -> base*o+b[k], s))); #select(isprime, s) }
%Y A329873 Cf. A007306, A036987, A078826, A303077.
%K A329873 nonn,look,base
%O A329873 0,6
%A A329873 _Rémy Sigrist_, Nov 23 2019