A335839 -id:A335839 - cp's OEIS frontend

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

Stanislav Sykora, May 10 2012

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

			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).

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

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 *)

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

Name edited by Michel Marcus, Sep 14 2020

a(7) from Thomas König, Jun 08 2021

A345296 Integers whose sum of digits in base b is the same for every prime b up to 17.

Original entry on oeis.org

0, 1, 70911040973874056146188543, 77332999599545910254098143, 139857575920160383360253101

Offset: 1

Thomas König, Jun 13 2021

This is a subset of A335839 for bases 2,3,5,11,13, which is a subset of A212222 for bases 2, 3, 5, 7, 11, which is a subset of A135127 for bases 2, 3, 5, 7, which is a subset of A135121 for bases 2, 3, 5, which is a subset of A037301 for bases 2, 3. The third term also occurs in A212223.

Based on a computer search, the next term is believed to be larger than 2.1e28. - Thomas König, Dec 08 2024

			77332999599545910254098143 = 11111111110111111001100100111011111011111110101111110010001010111101111101011011011111_2 =
1022220111022022121010102021222111100222120112011112120_3 = 10124120314223101043140143200022120033_5 = 3300561310042202241132326120022_7 = 7940063801000011830000282_11 = 1B101304100834600A304201_13 = 120802053643008116067_17. In these bases, the sum of digits is 63, so 77332999599545910254098143 is a term.

Cf. A212223, A212222, A135127, A135121, A037301, A335839.

a(5) from Thomas König, Dec 08 2024

cp's OEIS Frontend

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

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