A381487 Numbers which are a power of their digital root.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 81, 128, 256, 512, 729, 2401, 6561, 8192, 16384, 32768, 59049, 78125, 524288, 531441, 823543, 1048576, 2097152, 4782969, 33554432, 43046721, 67108864, 134217728, 282475249, 387420489, 1220703125, 2147483648, 3486784401, 4294967296

Offset

1,3

Links

Michel Marcus, Table of n, a(n) for n = 1..3347

Formula

a(n) = A381491(n)^A381492(n).

Example

a(12) = 128 is a term since 128 = 2^7 = A010888(128)^7.

Mathematica

A010888[n_]:=n - 9*Floor[(n-1)/9]; kmax=5*10^6; a={0, 1}; For[k=2, k<=kmax, k++, If[A010888[k]!=1, If[IntegerQ[Log[A010888[k], k]], AppendTo[a, k]]]]; a

Prog

(PARI) isok(k) = if ((k==0) || (k==1), return(1)); my(d=(k-1)%9+1); if (d>1, d^logint(k, d) == k); \\ Michel Marcus, Feb 26 2025
(PARI) lista(nn) = my(list = List()); listput(list, 0); listput(list, 1); for (n=2, 9, for (k=1, logint(nn, n), if ((n^k-1)%9+1 == n, listput(list, n^k)); ); ); vecsort(Vec(list)); \\ Michel Marcus, Feb 27 2025

Crossrefs

Cf. A010888, A023106, A381491, A381492.

Digital root of k^n: A000012 (1), A153130 (2), A100401 (3), A100402 (4), A070366 (5), A100403 (6), A070403 (7), A010689 (8), A010734 (9).

Sequence in context: A288946 A237346 A193757 * A246605 A038178 A023106

Adjacent sequences: A381484 A381485 A381486 * A381488 A381489 A381490

Keyword

nonn,base

Author

Stefano Spezia, Feb 25 2025

Status

approved