A379767 Triangle read by rows: row n lists numbers which are the n-th powers of their digit sum.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 81, 0, 1, 512, 4913, 5832, 17576, 19683, 0, 1, 2401, 234256, 390625, 614656, 1679616, 0, 1, 17210368, 52521875, 60466176, 205962976, 0, 1, 34012224, 8303765625, 24794911296, 68719476736, 0, 1, 612220032, 10460353203, 27512614111, 52523350144, 271818611107, 1174711139837, 2207984167552, 6722988818432

Offset

1,3

Comments

Each row begins with 0, 1. Solutions can have no more than R(n) digits, since (R(n)*9)^n < 10^R(n), hence, for each n, there are a finite number of solutions (Property 1 and table 1 of Clerc).

Links

René-Louis Clerc, Rows n=1...270 of triangle, flattened

René-Louis Clerc, Nombres de Niven-Harshad égaux à une puissance de la somme de leurs chiffres, zenodo.19037768, 2026.

Example

Triangle begins:
  1 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9;
  2 | 0, 1, 81;
  3 | 0, 1, 512, 4913, 5832, 17576, 19683;
  4 | 0, 1, 2401, 234256, 390625, 614656, 1679616;
  5 | 0, 1, 17210368, 52521875, 60466176, 205962976;
  6 | 0, 1, 34012224, 8303765625, 24794911296, 68719476736;
  7 | 0, 1, 612220032, 10460353203, 27512614111, 52523350144, 271818611107, 1174711139837, 2207984167552, 6722988818432;
  8 | 0, 1, 20047612231936, 72301961339136, 248155780267521;
  9 | 0, 1, 3904305912313344, 45848500718449031, 150094635296999121;
  ...

Prog

(PARI) R(n) = for(j=2, oo, if((j*9)^n <10^j, return(j)));
row(n) = my(L=List()); for (k=0, sqrtnint(10^R(n), n), if (k^n == sumdigits(k^n)^n, listput(L, k^n))); Vec(L)

Crossrefs

Rows 3..6 are A061209, A061210, A254000, A375343.

Row lengths are 1 + A046019(n).

Cf. A001014, A007953, A061211 (largest terms), A133509.

Cf. A152147.

Sequence in context: A309589 A309590 A169931 * A141022 A004720 A088118

Adjacent sequences: A379764 A379765 A379766 * A379768 A379769 A379770

Keyword

base,tabf,nonn

Author

René-Louis Clerc, Jan 02 2025

Status

approved