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A168036
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Difference between n' and n, where n' is the arithmetic derivative of n (A003415).
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22
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0, -1, -1, -2, 0, -4, -1, -6, 4, -3, -3, -10, 4, -12, -5, -7, 16, -16, 3, -18, 4, -11, -9, -22, 20, -15, -11, 0, 4, -28, 1, -30, 48, -19, -15, -23, 24, -36, -17, -23, 28, -40, -1, -42, 4, -6, -21, -46, 64, -35, -5, -31, 4, -52, 27, -39, 36, -35, -27, -58, 32, -60, -29
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OFFSET
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0,4
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COMMENTS
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Let k = n'-n. For k = -1 n is a primary pseudoperfect number (A054377), apart from n=1; For k=0 n is p^p, being p a prime number (A051674); For k = 1 n is a Giuga number (A007850).
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LINKS
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FORMULA
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Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = -1 + Sum_{p prime} 1/(p*(p-1)) = A136141 - 1 = -0.226843... . - Amiram Eldar, Dec 08 2023
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MAPLE
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with(numtheory);
local n, p;
for n from 0 to q do
print(n*add(op(2, p)/op(1, p), p=ifactors(n)[2])-n); od; end:
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MATHEMATICA
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np[k_] := Module[{f, n, m, p}, If[k < 2, np[k] = 0; Return[0], If[PrimeQ[k], np[k] = 1; Return[1], f = FactorInteger[k, 2]; m = f[[1, 1]]; n = k/m; p = m np[n] + n np[m]; np[k] = p; Return[p]]]];
Table[np[n] - n, {n, 0, 100}] (* Robert Price, Mar 14 2020 *)
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PROG
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(Haskell)
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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