A296341 - OEIS

A296341

Least number k such that the arithmetic derivatives of the composite numbers k-n and k+n are equal.

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

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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

Table of n, a(n) for n=1..46.

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 1 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