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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)). 4

%I

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

%N 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)).

%C 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

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

%p 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);

%p 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

%p 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

%t Select[Range[200],Total[IntegerDigits[#^Total[IntegerDigits[#]]]]==#&] (* _Harvey P. Dale_, Jul 26 2019 *)

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

%K easy,fini,nonn,full,base

%O 1,2

%A _Paolo P. Lava_ and _Giorgio Balzarotti_, Jan 22 2007

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Last modified October 21 19:29 EDT 2019. Contains 328315 sequences. (Running on oeis4.)