A212223 - OEIS

%I #40 Dec 26 2024 13:31:40

%S 2,6,6,882,1386,2007986541,70911040973874056146188543

%N a(n) is the least integer greater than 1 whose expansion in prime bases 2 to prime(n) have equal sum of digits.

%C Case a(1) is trivial since only base prime(1)=2 is involved.

%C Conjecture: the sequence never terminates.

%C a(7) > 2.3*10^16, if it exists. - _Giovanni Resta_, Oct 29 2018

%C Based on a search for the next term of A345296, a(8) is larger than 2.1*10^28. - _Thomas König_, Dec 15 2024

%e a(5) = 1386 because that number has the same sum of digits in the first 5 prime bases 2, 3, 5, 7, 11 (see A212222 and A000040).

%t f[n_] := Block[{p = Prime@ Range@ n, k = 2}, While[ Length[ Union[ Total@# & /@ IntegerDigits[k, p]]] != 1, k++]; k] (* _Robert G. Wilson v_, Oct 24 2014 *)

%o (PARI) isok(n, k) = my(s=hammingweight(k)); forprime (b=3, prime(n), if (sumdigits(k, b) != s, return (0))); return (1);

%o a(n) = my(k=2); while (!isok(n, k), k++); k; \\ _Michel Marcus_, Jun 08 2021

%Y Cf. A000040, A212222, A135127, A135121, A037301, A335839, A345296.

%K nonn,base,hard,more

%O 1,1

%A _Stanislav Sykora_, May 10 2012

%E Name edited by _Michel Marcus_, Sep 14 2020

%E a(7) from _Thomas König_, Jun 08 2021