A212223 a(n) is the least integer greater than 1 whose expansion in prime bases 2 to prime(n) have equal sum of digits.

2, 6, 6, 882, 1386, 2007986541, 70911040973874056146188543

Offset: 1

Author

Stanislav Sykora, May 10 2012

Keywords

• nonn
• base
• hard
• more

Comments

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

Conjecture: the sequence never terminates.

a(7) > 2.3*10^16, if it exists. - Giovanni Resta, Oct 29 2018

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

Examples

			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).
		

Crossrefs

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

Programs

Mathematica

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 *)

PARI

isok(n, k) = my(s=hammingweight(k)); forprime (b=3, prime(n), if (sumdigits(k, b) != s, return (0))); return (1);
a(n) = my(k=2); while (!isok(n, k), k++); k; \\ Michel Marcus, Jun 08 2021

Extensions

Name edited by Michel Marcus, Sep 14 2020

a(7) from Thomas König, Jun 08 2021