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 A296341 Least number k such that the arithmetic derivatives of the composite numbers k-n and k+n are equal. 0
 138004, 23, 2012, 136, 72708, 22, 1449858, 41, 264, 28, 1116, 107, 112, 44, 11752, 292, 1047798, 68, 88212, 71, 2478418, 54, 452, 119, 220, 92, 582, 592, 40284, 191, 329958, 89, 1600550, 602, 516798, 151, 2952, 140, 11434, 298, 125714, 212, 39654, 896, 822, 126 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS If the limitation of searching only for composite numbers k-n and k+n is removed, the terms we get are the average of two primes. LINKS EXAMPLE a(1) = 138004 because it is the least number k such that the composites k-1 and k+1 have arithmetic derivatives (k-1)' = (k+1)'. We see that (138004 - 1)' = (138004 + 1)' = 47351; a(2) = 23 because it is the least number k such that the composites k - 2 and k+2 have arithmetic derivatives (k-2)' = (k+2)'. We see that (23 - 1)' = (23 + 1). MAPLE with(numtheory): P:=proc(q) local a, h, n, p; for h from 2 to q do for n from h to q do if not isprime(n-h) and (n-h)*add(op(2, p)/op(1, p), p=ifactors(n-h)[2])= (n+h)*add(op(2, p)/op(1, p), p=ifactors(n+h)[2]) then print(n); break; fi; od; od; end: P(10^9); MATHEMATICA ad[n_] := With[{f = FactorInteger[n]}, n*Total[f[[All, 2]]/f[[All, 1]]]]; okQ[n_, k_] := If[Not[CompositeQ[k-n] && CompositeQ[k+n]], False, ad[k-n] == ad[k+n]]; a[n_] := For[k = 1, True, k++, If[okQ[n, k], Print["a(", n, ") = ", k]; Return[k]]]; Array[a, 46] (* Jean-François Alcover, Dec 20 2017 *) CROSSREFS Cf. A003415, A087711. Sequence in context: A234225 A110598 A069336 * A025301 A025320 A025293 Adjacent sequences:  A296338 A296339 A296340 * A296342 A296343 A296344 KEYWORD nonn,easy AUTHOR Paolo P. Lava, Dec 12 2017 STATUS approved

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Last modified July 31 20:24 EDT 2021. Contains 346377 sequences. (Running on oeis4.)