A061211 Largest number m such that m is the n-th power of the sum of its digits.

9, 81, 19683, 1679616, 205962976, 68719476736, 6722988818432, 248155780267521, 150094635296999121, 480682838924478847449, 23316389970546096340992, 2518170116818978404827136, 13695791164569918553628942336, 4219782742781494680756610809856

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

Cf. A061209, A061210, A046000, A076090, A046017.

Sequence in context: A053915 A067216 A076088 * A115988 A067506 A263816

Adjacent sequences: A061208 A061209 A061210 * A061212 A061213 A061214

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