A046000 a(n) is the largest number m equal to the sum of digits of m^n.

1, 9, 9, 27, 36, 46, 64, 68, 63, 81, 117, 108, 108, 146, 154, 199, 187, 216, 181, 207, 207, 225, 256, 271, 288, 337, 324, 307, 328, 341, 396, 443, 388, 423, 463, 477, 424, 495, 469, 523, 502, 432, 531, 572, 603, 523, 592, 666, 667, 695, 685, 685, 739, 746, 739, 683, 684, 802, 754, 845, 793, 833, 865

Offset

0,2

Comments

Cases a(n) = 1 begin: 0, 105, 164, 186, 194, 206, 216, 231, 254, 282, 285, ... Cf. A133509. - Jean-François Alcover, Jan 09 2018

It seems that a(n) < (5 + 2.7*log(n+1))*(n+1) for all n; the coefficient of the log can be decreased if the additive constant is increased. - M. F. Hasler, Jun 12 2026

The digit sum of m^n is at most 9*(1 + n log_10 m). Solving m = 9*(1 + n log_10 m) gives an upper bound for a(n). - M. F. Hasler, Jun 13 2026

References

Amarnath Murthy, The largest and the smallest m-th power whose digits sum /product is its m-th root. To appear in Smarandache Notions Journal, 2003.

Amarnath Murthy, e-book, "Ideas on Smarandache Notions" MS.LIT

Joe Roberts, "Lure of the Integers", The Mathematical Association of America, 1992, p. 172.

Links

T. D. Noe, Table of n, a(n) for n = 0..1000 (a(105), a(164), a(186), ... corrected by Mohammed Yaseen)

Formula

a(n) = A061211(n)^(1/n), for n > 0.

a(n) = A152147(n, A046019(n)). - M. F. Hasler, Jun 13 2026

Example

a(3) = 27 because 27 is the largest term of A046459, with 27^3 = 19683 = max(A061209) and 1+9+6+8+3 = 27.
a(5) = 46 because 46 is the largest term of A055575, with 46^5 = 205962976 = max(A254000) and 2+0+5+9+6+2+9+7+6 = 46.

Mathematica

meanDigit = 9/2; translate = 900; upperm[1] = translate;
upperm[n_] := Exp[-ProductLog[-1, -Log[10]/(meanDigit*n)]] + translate;
(* assuming that upper bound of m fits the implicit curve m = Log[10, m^n]*9/2 *)
a[0] = 1; a[n_] := (For[max = m = 0, m <= upperm[n], m++, If[m == Total[IntegerDigits[m^n]], max = m]]; max);
Table[a[n], {n, 0, 1000}] (* Jean-François Alcover, Jan 09 2018, updated Jul 07 2022 *)

Prog

(Python)
def ok(k, n): return sum(map(int, str(k**n))) == k
def a(n):
    d, lim = 1, 1
    while lim < n*9*d: d, lim = d+1, lim*10
    return next(k for k in range(lim, 0, -1) if ok(k, n))
print([a(n) for n in range(63)]) # Michael S. Branicky, Jul 06 2022

Crossrefs

Cf. A046459, A055575, A055576, A055577.

Cf. A046017, A046019, A046471, A133509, A152147 (table with solutions for all n).

Cf. A061211, A076090.

Sequence in context: A339716 A111005 A076089 * A147378 A146604 A294691

Adjacent sequences: A045997 A045998 A045999 * A046001 A046002 A046003

Keyword

base,nonn,nice

Author

David W. Wilson and Patrick De Geest

Extensions

More terms from Asher Auel, Jun 01 2000

More terms from Franklin T. Adams-Watters, Sep 01 2006

Edited by N. J. A. Sloane at the suggestion of David Wasserman, Dec 12 2007

Status

approved