A125526 Numbers k for which the sum of the digits of k raised to the sum of the digits of k itself is equal to k. If "sumdigit" denotes the sum of the digits of a number then these are the numbers k such that k = sumdigit(k^sumdigit(k)).

1, 22, 34, 43, 54, 81, 82, 169, 187

Offset

1,2

Comments

There are no other terms. Proof: Assume the next term has d digits. 10^d > k >= 10^(d-1); sumdigit(k) >= 9d; k^sumdigit(k) < (10^d)^(9d) < 10^(9d^2); 9*(9d^2+1) > sumdigit(k^sumdigit(k)); 9*(9d^2+1) > k 9*(9d^2+1) > 10^(d-1). So d < 5. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Mar 11 2007

Links

Table of n, a(n) for n=1..9.

Example

a(2)=22 because 2 + 2 = 4, 22^4 = 234256, 2 + 3 + 4 + 2 + 5 + 6 = 22.

Maple

P:=proc(n) local i, j, k, w; for i from 1 by 1 to n do w:=0; k:=i; while k>0 do w:=w+k-trunc(k/10)*10; k:=trunc(k/10); od; k:=i^w; w:=0; while k>0 do w:=w+k-trunc(k/10)*10; k:=trunc(k/10); od; if (i=w) then print(w); fi; od; end: P(200);
sod := proc(n, b) convert(convert(n, base, b), `+`) end; b:=10: L:=[]: for w to 1 do for n from 1 to 10^3 do x:=sod(n^sod(n, b), b); if x=n then print(n); L:=[op(L), n]; fi; od od; L; # Walter Kehowski, Feb 12 2007
sd:=proc(n) local nn: nn:=convert(n, base, 10): sum(nn[j], j=1..nops(nn)) end: a:=proc(n) if sd(n^sd(n))=n then n else fi end: seq(a(n), n=1..500); # Emeric Deutsch, Feb 16 2007

Mathematica

Select[Range[200], Total[IntegerDigits[#^Total[IntegerDigits[#]]]]==#&] (* Harvey P. Dale, Jul 26 2019 *)

Crossrefs

Cf. A124053, A124359, A124360, A046019, A124365, A124366, A124367.

Sequence in context: A213974 A103320 A306882 * A181177 A124317 A254691

Adjacent sequences: A125523 A125524 A125525 * A125527 A125528 A125529

Keyword

easy,fini,nonn,full,base

Author

Paolo P. Lava and Giorgio Balzarotti, Jan 22 2007

Status

approved