A269719 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, 1728787, 1900523, 2045171

Offset

1,1

Links

Table of n, a(n) for n=1..37.

Paolo P. Lava, Terms of the sequence and their fixed power

Example

3^0 = 1 and 3' = 1;
4^1 = 4 and 4' = 4;
1^3 + 4^3 + 2^3 = 73 and 143' = 73.

Maple

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);

Mathematica

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 *)

Crossrefs

Cf. A003415.

Sequence in context: A035359 A365577 A325410 * A214626 A143593 A364089

Adjacent sequences: A269716 A269717 A269718 * A269720 A269721 A269722

Keyword

nonn,easy

Author

Paolo P. Lava, Mar 04 2016

Status

approved