A272780 Numbers k such that k' = concat(s,t) and k = s*t, where k' is the arithmetic derivative of k.

648, 750, 4050, 35574, 50000, 85440, 87500, 105984, 183141, 193750, 213759, 223600, 290625, 297500, 332000, 368550, 468750, 473000, 479600, 484375, 511584, 520625, 529984, 531050, 678125, 754392, 825000, 987500, 999840, 1011375, 1054614, 1065625, 1112625, 1139050

Offset

1,1

Links

Jinyuan Wang, Table of n, a(n) for n = 1..300

Example

648' = 1836 and 18*36 = 648;
750' = 1075 and 10*75 = 750.

Maple

with(numtheory): P:=proc(q) local a, b, c, i, n, p;
for n from 1 to q do c:= n*add(op(2, p)/op(1, p), p=ifactors(n)[2]);
for i from 1 to ilog10(c) do a:=trunc(c/10^i);  b:=c-a*10^i;
if a+b=n then print(n); break; fi; od; od; end: P(10^9);

Mathematica

Select[Range[10^5], Function[n, Total@ Boole@ Function[k, And[k == FromDigits[Join[IntegerDigits@ First@ #, IntegerDigits@ Last@ #]], n == First@ # Last@ #] & /@ Map[FromDigits /@ TakeDrop[IntegerDigits@ k, #] &, Range[IntegerLength@ k - 1]]][If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger@ Abs@ n]]] > 0]] (* Michael De Vlieger, May 07 2016, Version 10.2 *)
ok[t_, n_] := Catch@ Block[{p=10}, While[p < t, If[n == Mod[t, p] Floor[ t/p], Throw@True, p *= 10]]; False]; Select[Range[2, 10^5], ok[# Plus @@ (#2/#1 & @@@ FactorInteger@#), #] &] (* Giovanni Resta, May 07 2016, older Mma *)

Crossrefs

Cf. A003415.

Sequence in context: A271639 A034281 A157432 * A171606 A006914 A337313

Adjacent sequences: A272777 A272778 A272779 * A272781 A272782 A272783

Keyword

nonn,base

Author

Paolo P. Lava, May 06 2016

Status

approved