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

%I A379749 #15 Jan 04 2025 22:30:19
%S A379749 5,7,13,41,31,43,113,73,181,331,397,157,547,211,241,1361,307,2053,761,
%T A379749 421,463,1013,1657,601,1301,3511,757,2437,1741,1861,5953,2113,1123,
%U A379749 2381,2521,6661,4219,1483,3121,13121,1723,3613,9461,9901,6211,12973,4513,7057,7351,2551,15913,8269,25759,2971
%N A379749 a(n) is the first prime that has digit sum n in base n and n+1 in base n+1.
%C A379749 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.
%H A379749 Robert Israel, <a href="/A379749/b379749.txt">Table of n, a(n) for n = 2..1000</a>
%e A379749 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.
%p A379749 f:= proc(n) local k,v,x;
%p A379749   for k from 1 do
%p A379749     v:= convert(convert(k,base,n),`+`);
%p A379749     if v <= n then
%p A379749       x:= k*n + n-v;
%p A379749       if convert(convert(x,base,n+1),`+`) = n+1 and isprime(x) then return x fi
%p A379749     fi
%p A379749   od;
%p A379749 end proc:
%p A379749 map(f, [$2 .. 100]);
%t A379749 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 *)
%o A379749 (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
%o A379749 (Python)
%o A379749 from sympy import isprime
%o A379749 from sympy.ntheory import digits
%o A379749 def nextsod(n, base):
%o A379749     c, b, w = 0, base, 0
%o A379749     while True:
%o A379749         d = n%b
%o A379749         if d+1 < b and c:
%o A379749             return (n+1)*b**w + ((c-1)%(b-1)+1)*b**((c-1)//(b-1))-1
%o A379749         c += d; n //= b; w += 1
%o A379749 def A226636gen(sod=3, base=3): # generator of terms for any sod, base
%o A379749     an = (sod%(base-1)+1)*base**(sod//(base-1))-1
%o A379749     while True: yield an; an = nextsod(an, base)
%o A379749 def a(n):
%o A379749     for k in A226636gen(sod=n, base=n):
%o A379749         if sum(digits(k, n+1)[1:]) == n+1 and isprime(k):
%o A379749             return k
%o A379749 print([a(n) for n in range(2, 56)]) # _Michael S. Branicky_, Jan 04 2025
%Y A379749 Cf. A002061, A055494, A379743.
%K A379749 nonn,base,look
%O A379749 2,1
%A A379749 _Robert Israel_, Jan 01 2025