

A296341


Least number k such that the arithmetic derivatives of the composite numbers kn 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
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OFFSET

1,1


COMMENTS

If the limitation of searching only for composite numbers kn 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 k1 and k+1 have arithmetic derivatives (k1)' = (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 (k2)' = (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(nh) and
(nh)*add(op(2, p)/op(1, p), p=ifactors(nh)[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[kn] && CompositeQ[k+n]], False, ad[kn] == ad[k+n]];
a[n_] := For[k = 1, True, k++, If[okQ[n, k], Print["a(", n, ") = ", k]; Return[k]]];
Array[a, 46] (* JeanFrançois 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



