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Search: id:A099305
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| A099305 |
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Number of solutions of the equation (n+k)' = n' + k', with 1 <= k <= 2n, where n' denotes the arithmetic derivative of n. |
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3
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| 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 1, 2, 3, 3, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 1, 2, 1, 3, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 2, 2, 1, 3, 3, 3, 1, 2, 2, 3, 2
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Observe that when n and c*n have the same parity, a(c*n) >= a(n) for all integers c. For even n, there are always at least two solutions, k=n/2 and k=2n. For odd n, k=2n is always a solution.
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REFERENCES
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See A003415
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MATHEMATICA
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dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; Table[lst={}; k=0; While[k<2n, k++; While[k<=2n && dn[n]+dn[k] != dn[n+k], k++ ]; If[dn[n]+dn[k]==dn[n+k], AppendTo[lst, k]]]; Length[lst], {n, 100}]
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CROSSREFS
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Cf. A003415 (arithmetic derivative of n), A099304 (least k > 0 such that (n+k)' = n' + k').
Sequence in context: A058061 A064547 A086435 this_sequence A033109 A111627 A008618
Adjacent sequences: A099302 A099303 A099304 this_sequence A099306 A099307 A099308
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Oct 12 2004
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