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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

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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);