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

%I A046000 #40 Oct 27 2023 22:05:37
%S A046000 1,9,9,27,36,46,64,68,63,81,117,108,108,146,154,199,187,216,181,207,
%T A046000 207,225,256,271,288,337,324,307,328,341,396,443,388,423,463,477,424,
%U A046000 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
%N A046000 a(n) is the largest number m equal to the sum of digits of m^n.
%C A046000 Cases a(n) = 1 begin: 0, 105, 164, 186, 194, 206, 216, 231, 254, 282, 285, ... Cf. A133509. - _Jean-François Alcover_, Jan 09 2018
%D A046000 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.
%D A046000 Amarnath Murthy, e-book, "Ideas on Smarandache Notions" MS.LIT
%D A046000 Joe Roberts, "Lure of the Integers", The Mathematical Association of America, 1992, p. 172.
%H A046000 T. D. Noe, <a href="/A046000/b046000.txt">Table of n, a(n) for n = 0..1000</a> (a(105), a(164), a(186), ... corrected by Mohammed Yaseen)
%F A046000 a(n) = A061211(n)^(1/n), for n > 0.
%e A046000 a(3) = 27 because 27 is the largest number with 27^3 = 19683 and 1+9+6+8+3 = 27.
%e A046000 a(5) = 46 because 46 is the largest number with 46^5 = 205962976 and 2+0+5+9+6+2+9+7+6 = 46.
%t A046000 meanDigit = 9/2; translate = 900; upperm[1] = translate;
%t A046000 upperm[n_] := Exp[-ProductLog[-1, -Log[10]/(meanDigit*n)]] + translate;
%t A046000 (* assuming that upper bound of m fits the implicit curve m = Log[10, m^n]*9/2 *)
%t A046000 a[0] = 1; a[n_] := (For[max = m = 0, m <= upperm[n], m++, If[m == Total[IntegerDigits[m^n]], max = m]]; max);
%t A046000 Table[a[n], {n, 0, 1000}] (* _Jean-François Alcover_, Jan 09 2018, updated Jul 07 2022 *)
%o A046000 (Python)
%o A046000 def ok(k, n): return sum(map(int, str(k**n))) == k
%o A046000 def a(n):
%o A046000     d, lim = 1, 1
%o A046000     while lim < n*9*d: d, lim = d+1, lim*10
%o A046000     return next(k for k in range(lim, 0, -1) if ok(k, n))
%o A046000 print([a(n) for n in range(63)]) # _Michael S. Branicky_, Jul 06 2022
%Y A046000 Cf. A046459, A055575, A055576, A055577.
%Y A046000 Cf. A046017, A046019, A046471, A133509, A152147.
%Y A046000 Cf. A061211, A076090.
%K A046000 base,nonn,nice
%O A046000 0,2
%A A046000 David W. Wilson and _Patrick De Geest_
%E A046000 More terms from _Asher Auel_, Jun 01 2000
%E A046000 More terms from _Franklin T. Adams-Watters_, Sep 01 2006
%E A046000 Edited by _N. J. A. Sloane_ at the suggestion of _David Wasserman_, Dec 12 2007