A384211 a(n) is the number of distinct ways of representing n in any integer base >= 2 using only prime digits.

0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 2, 2, 2, 1, 4, 2, 3, 2, 3, 1, 6, 2, 3, 4, 4, 1, 6, 2, 5, 4, 5, 2, 7, 2, 7, 4, 5, 3, 8, 4, 9, 3, 7, 3, 12, 3, 6, 5, 6, 4, 11, 2, 9, 4, 9, 6, 13, 3, 11, 8, 12, 3, 12, 3, 13, 7, 8, 5, 14, 5, 13, 5, 11, 4, 15, 3, 13, 8, 10, 7, 15

Offset: 0

Author

Felix Huber, May 23 2025

Keywords

• nonn
• base

Comments

The representations of n remain the same for bases greater than n, as they all consist solely of the digit n.

Examples

			The a(43) = 9 representations of [4,3] in base 10 using only prime digits are [2,2,3] in base 4, [5,3] in base 8, [3,7] in base 12, [2,13] in base 15, [2,11] in base 16, [2,7] in base 18, [2,5] in base 19, [2,3] in base 20 and [43] in bases >= 44.
		

Links

• Felix Huber, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A055240, A355034.

Programs

Maple

A384211:=proc(n)
    local a,b,c;
    a:=0;
    for b from 2 to n+1 do
        c:=convert(n,'base',b);
        if select(isprime,c)=c then
            a:=a+1
        fi
    od;
    return a
end proc;
seq(A384211(n),n=0..87);
A384211representations:=proc(n)
    local L,b,c;
    L:=[];
    for b from 2 to n+1 do
        c:=convert(n,'base',b);
        if select(isprime,c)=c then
            L:=[op(L),b,ListTools:-Reverse(c)]
        fi
    od;
    return op(L)
end proc;
A384211representations(43);

Mathematica

a[n_] := Boole[PrimeQ[n]] + Count[Range[2, n-1], ?(AllTrue[IntegerDigits[n, #], PrimeQ] &)]; Array[a, 100 ,0] (* _Amiram Eldar, May 23 2025 *)

PARI

a(n) = sum(k=2, n+1, my(d=digits(n, k)); #select(isprime, d) == #d); \\ Michel Marcus, May 26 2025

Python

from sympy import isprime
from sympy.ntheory import digits
def a(n): return len(set(t for b in range(2, n+2) if all(map(isprime, (t:=tuple(digits(n, b)[1:]))))))
print([a(n) for n in range(84)]) # Michael S. Branicky, May 23 2025