A061210 -id:A061210 - cp's OEIS frontend

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

René-Louis Clerc, Jan 02 2025

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).

			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;
  ...

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, 2024.

Rows 3..6 are A061209, A061210, A254000, A375343.
Row lengths are 1 + A046019(n).
Cf. A001014, A007953, A061211 (largest terms), A133509.
Cf. A152147.

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)

A370250 Numbers k such that the sum of the digits times the square of the sum of the fourth powers of the digits equals k.

Original entry on oeis.org

0, 1, 5873656512, 7253758561, 29961747275

Offset: 1

René-Louis Clerc, Feb 13 2024

There are exactly 5 such numbers (Property 17 of Clerc).

			7253758561 = (7+2+5+3+7+5+8+5+6+1)*(7^4 + 2^4 + 5^4 + 3^4 + 7^4 + 5^4 + 8^4 + 5^4 + 6^4 + 1^4)^2 = 49*148035889 = 7253758561.

René-Louis Clerc, Quelques nombres de Niven-Harshad particuliers, 2023.

Cf. A368939, A115518, A257766, A061209, A061210, A254000, A130680, A366507, A366512.

niven142(k) = my(d=digits(k)); vecsum(d)*sum(i=1, #d, d[i]^4)^2 == k;
for(k=0,10^12,if(niven142(k)==1,print1(k, ", ")))

cp's OEIS Frontend

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

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