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A240053 3rd Arithmetic derivation of products of 2 successive prime numbers (A006094).

%I A240053 #24 Apr 15 2014 02:50:08
%S A240053 0,16,32,10,48,1,92,1,92,96,156,1,128,44,188,608,248,1408,22,1472,240,
%T A240053 324,368,30,86,288,32,1188,1,1552,30,560,476,2176,924,476,5120,60,432,
%U A240053 2176,1148,512,4480,1,1300,324,1,391,1052,46,720,3232,672,2304,1448,860,2484,1036,226,768,7232,1628
%N A240053 3rd Arithmetic derivation of products of 2 successive prime numbers (A006094).
%C A240053 The first arithmetic derivation of products of 2 successive prime numbers (A006094) is the sum of 2 successive prime numbers (A001043). A001043 = (A006094)’. The second arithmetic derivation is (A240052) = (A001043)’ = (A006094)’’. The third arithmetic derivation of products of 2 successive prime numbers (A006094) is a(n) = (A240052)’ = (A001043)’’ = (A006094)’’’.
%H A240053 Freimut Marschner, <a href="/A240053/b240053.txt">Table of n, a(n) for n = 1..429</a>
%H A240053 Wikipedia, <a href="http://en.wikipedia.org/wiki/Arithmetic_derivative">Arithmetic derivative</a>
%H A240053 Wikipedia, <a href="http://en.wikipedia.org/wiki/P-derivation">p-derivation</a>
%F A240053 a(n) = (A006094(n))’’’.
%F A240053 a(n) = A099306(A006094(n)).
%F A240053 a(n) = A003415(A240052(n)).
%e A240053 a(12)=(A006094(12))'''=(37*41)'''=(A001043(12))''=(78)''=(71)'=1;
%e A240053 a(14)=(A006094(14))'''=(43*47)'''=(A001043(12))''=(90)''=(123)'=44.
%p A240053 with(numtheory); P:= proc(q) local a,b,c,d,n,p;  a:=ithprime(n)*ithprime(n+1);
%p A240053 for n from 1 to q do a:=ithprime(n)*ithprime(n+1);
%p A240053 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 A240053 d:=c*add(op(2,p)/op(1,p),p=ifactors(c)[2]); print(d);
%p A240053 od; end: P(10^4); # _Paolo P. Lava_, Apr 07 2014
%Y A240053 Cf. A006094, A001043, A240052.
%Y A240053 Cf. A003415 (1st derivative), A068346 (2nd derivative), A099306 (3rd derivative).
%K A240053 nonn
%O A240053 1,2
%A A240053 _Freimut Marschner_, Mar 31 2014