A379749 a(n) is the first prime that has digit sum n in base n and n+1 in base n+1.

5, 7, 13, 41, 31, 43, 113, 73, 181, 331, 397, 157, 547, 211, 241, 1361, 307, 2053, 761, 421, 463, 1013, 1657, 601, 1301, 3511, 757, 2437, 1741, 1861, 5953, 2113, 1123, 2381, 2521, 6661, 4219, 1483, 3121, 13121, 1723, 3613, 9461, 9901, 6211, 12973, 4513, 7057, 7351, 2551, 15913, 8269, 25759, 2971

Offset: 2

Author

Robert Israel, Jan 01 2025

Keywords

• nonn
• base
• look

Comments

For n >= 3, the least number with digit sum n in base n and n+1 in base n+1 is A002061(n) = n^2 - n + 1. This is prime for n in A055494. Thus (if it exists) a(n) >= A002061(n), with equality for n in A055494.

Examples

			For n = 8, a(8) = 113 because 113 is prime, 113 = 161_8 = 135_9 has digit sums 8 in base 8 and 9 in base 9, and no smaller prime works.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..1000

Crossrefs

Cf. A002061, A055494, A379743.

Programs

Maple

f:= proc(n) local k,v,x;
  for k from 1 do
    v:= convert(convert(k,base,n),`+`);
    if v <= n then
      x:= k*n + n-v;
      if convert(convert(x,base,n+1),`+`) = n+1 and isprime(x) then return x fi
    fi
  od;
end proc:
map(f, [$2 .. 100]);

Mathematica

a[n_]:=Module[{k=1}, While[DigitSum[Prime[k],n]!=n || DigitSum[Prime[k],n+1]!=n+1, k++]; Prime[k]]; Array[a,54,2] (* Stefano Spezia, Jan 01 2025 *)

PARI

a(n) = my(p=2); while ((sumdigits(p, n) != n) || (sumdigits(p, n+1) != n+1), p=nextprime(p+1)); p; \\ Michel Marcus, Jan 02 2025

Python

from sympy import isprime
from sympy.ntheory import digits
def nextsod(n, base):
    c, b, w = 0, base, 0
    while True:
        d = n%b
        if d+1 < b and c:
            return (n+1)*b**w + ((c-1)%(b-1)+1)*b**((c-1)//(b-1))-1
        c += d; n //= b; w += 1
def A226636gen(sod=3, base=3): # generator of terms for any sod, base
    an = (sod%(base-1)+1)*base**(sod//(base-1))-1
    while True: yield an; an = nextsod(an, base)
def a(n):
    for k in A226636gen(sod=n, base=n):
        if sum(digits(k, n+1)[1:]) == n+1 and isprime(k):
            return k
print([a(n) for n in range(2, 56)]) # Michael S. Branicky, Jan 04 2025