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A272780
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Numbers k such that k' = concat(s,t) and k = s*t, where k' is the arithmetic derivative of k.
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1
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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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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648' = 1836 and 18*36 = 648;
750' = 1075 and 10*75 = 750.
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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