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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.
2
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, 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, 4, 9, 7, 9, 3, 11, 9, 11, 6, 11, 7, 10, 3, 8, 8, 12, 6
OFFSET
0,6
COMMENTS
This sequence is unbounded.
LINKS
FORMULA
A078826(n) <= a(n) <= A007306(n+1).
a(2*n) = a(n) + A036987(n) for any n > 0.
a(2^n) = 1 for any n > 0.
EXAMPLE
The first terms, alongside the binary representations of n and of the corresponding prime numbers, are:
n a(n) bin(n) {bin(p)}
-- ---- ------ --------------------
0 0 0 {}
1 0 1 {}
2 1 10 {10}
3 1 11 {11}
4 1 100 {10}
5 3 101 {10, 11, 101}
6 2 110 {10, 11}
7 2 111 {11, 111}
8 1 1000 {10}
9 3 1001 {10, 11, 101}
10 3 1010 {10, 11, 101}
11 5 1011 {10, 11, 101, 111, 1011}
12 2 1100 {10, 11}
MAPLE
b:= proc(n) option remember; `if`(n=0, {0},
map(x-> [x, 2*x+r][], b(iquo(n, 2, 'r'))))
end:
a:= n-> nops(select(isprime, b(n))):
seq(a(n), n=0..84); # Alois P. Heinz, Jan 26 2022
PROG
(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) }
CROSSREFS
KEYWORD
nonn,look,base
AUTHOR
Rémy Sigrist, Nov 23 2019
STATUS
approved