OFFSET
1,1
COMMENTS
Clearly m = 1 always works, so a(n) exists for all n. - Farideh Firoozbakht, Nov 23 2007
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
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.
Amarnath Murthy, e-book, "Ideas on Smarandache Notions", manuscript.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..105
EXAMPLE
a(3) = 19683 = 27^3 and no bigger number can have this property. (This has been established in the Murthy reference.)
a(4) = 1679616 = (1+6+7+9+6+1+6)^4 = 36^4.
MATHEMATICA
meanDigit = 9/2; translate = 900; upperm[1] = translate;
upperm[n_] := Exp[-ProductLog[-1, -Log[10]/(meanDigit*n)]] + translate;
a[n_] := (For[max = m = 1, m <= upperm[n], m++, If[m == Total[ IntegerDigits[ m^n ] ], max = m]]; max^n);
Array[a, 14] (* Jean-François Alcover, Jan 09 2018 *)
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Amarnath Murthy, Apr 21 2001
EXTENSIONS
More terms from Ulrich Schimke, Feb 11 2002
Edited by N. J. A. Sloane at the suggestion of Farideh Firoozbakht, Dec 04 2007
STATUS
approved