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%I #18 Jan 09 2018 03:08:33
%S 9,81,19683,1679616,205962976,68719476736,6722988818432,
%T 248155780267521,150094635296999121,480682838924478847449,
%U 23316389970546096340992,2518170116818978404827136,13695791164569918553628942336,4219782742781494680756610809856
%N Largest number m such that m is the n-th power of the sum of its digits.
%C Clearly m = 1 always works, so a(n) exists for all n. - _Farideh Firoozbakht_, Nov 23 2007
%C 105 is the smallest number n such that a(n)=1. This means that if n<105 there exists at least one number m greater than 1 such that m is the n-th power of the sum of its digits while 1 is the only number m such that m is the 105th power of the sum of its digits. A133509 gives n such that a(n) = 1. - _Farideh Firoozbakht_, Nov 23 2007
%D 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.
%D Amarnath Murthy, e-book, "Ideas on Smarandache Notions", manuscript.
%H T. D. Noe, <a href="/A061211/b061211.txt">Table of n, a(n) for n = 1..105</a>
%e a(3) = 19683 = 27^3 and no bigger number can have this property. (This has been established in the Murthy reference.)
%e a(4) = 1679616 = (1+6+7+9+6+1+6)^4 = 36^4.
%t meanDigit = 9/2; translate = 900; upperm[1] = translate;
%t upperm[n_] := Exp[-ProductLog[-1, -Log[10]/(meanDigit*n)]] + translate;
%t a[n_] := (For[max = m = 1, m <= upperm[n], m++, If[m == Total[ IntegerDigits[ m^n ] ], max = m]]; max^n);
%t Array[a, 14] (* _Jean-François Alcover_, Jan 09 2018 *)
%Y Cf. A061209, A061210, A046000, A076090, A046017.
%K nonn,base
%O 1,1
%A _Amarnath Murthy_, Apr 21 2001
%E More terms from Ulrich Schimke, Feb 11 2002
%E Edited by _N. J. A. Sloane_ at the suggestion of _Farideh Firoozbakht_, Dec 04 2007
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