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A349484
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Niven numbers whose arithmetic derivative is also a Niven number (A005349).
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1
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2, 3, 4, 5, 6, 7, 8, 9, 10, 18, 20, 21, 27, 36, 48, 50, 54, 72, 81, 100, 108, 111, 112, 135, 153, 156, 180, 192, 201, 209, 210, 216, 224, 225, 230, 243, 280, 288, 306, 324, 336, 351, 364, 378, 392, 400, 405, 407, 420, 432, 441, 480, 481, 486, 500, 504, 511, 512
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OFFSET
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1,1
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COMMENTS
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The sequence is infinite because the numbers of the form m = 2*10^(10^k), k >= 1, are terms. Indeed, m is a Niven number, m' = 10^(10^k) + 2*10^k*10^(10^k - 1)*7 = 10^(10^k - 1)*(10 + 140*10^k) = 10^(10^k)*(1 + 14*10^k), digsum(m') = 6 and m' is divisible by 6, so it is a Niven number.
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LINKS
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EXAMPLE
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MATHEMATICA
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nivenQ[n_] := Divisible[n, Plus @@ IntegerDigits[n]]; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[2, 512], And @@ nivenQ /@ {#, d[#]} &] (* Amiram Eldar, Nov 20 2021 *)
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PROG
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(Magma) f:=func<n |n le 1 select 0 else n*(&+[Factorisation(n)[i][2] / Factorisation(n)[i][1]: i in [1..#Factorisation(n)]])>; a:=[]; niven:=func<n|n mod &+Intseq(n) eq 0>; [n:n in [2..520]|niven(n) and niven(Floor(f(n)))];
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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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