A329873 - OEIS

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.

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

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OFFSET

0,6

COMMENTS

This sequence is unbounded.

LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..16384

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

Cf. A007306, A036987, A078826, A303077.

Sequence in context: A171900 A204257 A074976 * A068448 A054081 A164585

Adjacent sequences: A329870 A329871 A329872 * A329874 A329875 A329876

KEYWORD

nonn,look,base

AUTHOR

Rémy Sigrist, Nov 23 2019

STATUS

approved