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A240054 4th arithmetic derivative of products of 2 successive prime numbers (A006094).

%I A240054 #26 Feb 03 2019 03:27:16
%S A240054 0,32,80,7,112,0,96,0,96,272,220,0,448,48,192,1552,380,5056,13,4480,
%T A240054 608,756,752,31,45,912,80,2484,0,3120,31,1312,572,7744,1448,572,26624,
%U A240054 92,1296,7744,1340,2304,17216,0,1920,756,0,40,1056,25,2064,8112,2000,10752,2180,1052,5076,1212,115,3328,21760,1820
%N A240054 4th arithmetic derivative of products of 2 successive prime numbers (A006094).
%C A240054 Let a'=a1 be the first arithmetic derivative, then a2 is the second and so on. It is interesting to examine the length of successive arithmetic derivatives ending with 1. For example, a(168) = 445 is the 4th arithmetic derivative of prime(168)*prime(169) = 997*1009 = 1005973. The example given here is of length 11; that means that the 11th arithmetic derivative of 1005973 is 1.
%H A240054 Freimut Marschner, <a href="/A240054/b240054.txt">Table of n, a(n) for n = 1..429</a>
%H A240054 Wikipedia, <a href="http://en.wikipedia.org/wiki/Arithmetic_derivative">Arithmetic derivative</a>
%H A240054 Wikipedia, <a href="http://en.wikipedia.org/wiki/P-derivation">p-derivation</a>
%F A240054 a(n) = (A006094(n))''''.
%F A240054 a(n) = A068346(A068346(n)) = A003415(A003415(A003415(A003415(n)))).
%e A240054 (997*1009)' = a, a' = a1 = 2006, a2 = 1155, a3 = 886, a4 = 445, a5 = 94, a6 = 49, a7 = 14, a8 = 9, a9 = 6, a10 = 5, a11 = 1.
%p A240054 with(numtheory); P:= proc(q) local a,b,c,d,f,n,p;  a:=ithprime(n)*ithprime(n+1);
%p A240054 for n from 1 to q do a:=ithprime(n)*ithprime(n+1);
%p A240054 b:=a*add(op(2,p)/op(1,p),p=ifactors(a)[2]); c:=b*add(op(2,p)/op(1,p),p=ifactors(b)[2]);
%p A240054 d:=c*add(op(2,p)/op(1,p),p=ifactors(c)[2]); f:=d*add(op(2,p)/op(1,p),p=ifactors(d)[2]);
%p A240054 print(d); od; end: P(10^4); # _Paolo P. Lava_, Apr 07 2014
%Y A240054 Cf. A006094, A001043, A068346, A003415, A240052, A240053.
%K A240054 nonn
%O A240054 1,2
%A A240054 _Freimut Marschner_, Mar 31 2014