proposed

approved

editing

proposed

f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger@ Abs@ n]]; Select[Range[3, 10^5], Function[k, IntegerQ@ SelectFirst[Range[0, 10], Function[d, If[MemberQ[d, 0] && # == 0, Total@ Power[d /. 0 -> Nothing, #] == f@ k, Total@ Power[d, #] == f@ k]]@ IntegerDigits@ k &]]] (* Michael De Vlieger, Mar 04 2016, Version 10, f(n) after Michael Somos at A003415 *)

proposed

editing

editing

proposed

3, 4, 5, 7, 142, 581, 6127, 8549, 12643, 16999, 51703, 57121, 86833, 89195, 92029, 103039, 104647, 112093, 137317, 149851, 218269, 261883, 266923, 323723, 336273, 449881, 505891, 524371, 610171, 617569, 907873, 999643, 1119253, 1134227, 1728787, 1900523, 2045171

Paolo P. Lava, <a href="/A269719/a269719_1.txt">Terms of the sequence and their fixed power</a>

Paolo P. Lava, <a href="/A269719/a269719.txt">Terms of the sequence and their fixed power</a>

allocated for Paolo P. Lava

Numbers whose arithmetic derivative is equal to the sum of some fixed power of their digits.

3, 4, 5, 7, 142, 581, 6127, 8549, 12643, 16999, 51703, 57121, 86833, 89195, 92029, 103039, 104647, 112093, 137317, 149851, 218269, 261883, 266923, 323723, 336273, 449881, 505891, 524371, 610171, 617569, 907873, 999643, 1119253, 1134227

1,1

3^0 = 1 and 3' = 1;

4^1 = 4 and 4' = 4;

1^3 + 4^3 + 2^3 = 73 and 143' = 73.

with(numtheory): P:= proc(q) local a, b, c, d, j, k, n, ok; for n from 3 to q do a:=[]; b:=n; ok:=0;

d:=n*add(op(2, p)/op(1, p), p=ifactors(n)[2]); a:=[]; b:=n; ok:=0;

for k from 1 to ilog10(n)+1 do if (b mod 10)>1 then ok:=1; fi; a:=[(b mod 10), op(a)]; b:=trunc(b/10); od; b:=-1; c:=0;

if ok=1 then while c<d do b:=b+1;

if b>0 then c:=add(a[k]^b, k=1..nops(a)); else for k from 1 to nops(a) do if a[k]=0 then c:=0; break;

else c:=c+1; fi; od; fi; od; if c=d then lprint(n, b); fi; fi; od; end: P(10^9);

Cf. A003415.

allocated

nonn,easy

Paolo P. Lava, Mar 04 2016

approved

editing

allocated for Paolo P. Lava

allocated

approved