A240052 - OEIS

%I #25 Apr 16 2014 05:52:32

%S 1,12,16,21,44,31,60,41,56,92,72,71,124,123,140,240,244,448,121,384,

%T 236,297,176,161,249,284,247,540,191,608,221,272,380,912,520,380,1024,

%U 371,428,912,852,508,1472,433,696,297,293,705,860,493,716,1456,668,512,924,636,1188,552,669,764,2112,1340,521,1504,951,1836,672,1176,1300,1107,1076,737,908,1520,641,776,661,821,1647,1416,1828

%N 2nd arithmetic derivative of products of 2 successive prime numbers (A006094).

%C The first arithmetic derivative of products of 2 successive prime numbers (A006094) is the sum of 2 successive prime numbers (A001043). A001043 = (A006094)’. The second arithmetic derivative is a(n)=( A001043)’ = (A006094)’’.

%H Freimut Marschner, <a href="/A240052/b240052.txt">Table of n, a(n) for n = 1..429</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Arithmetic_derivative">Arithmetic derivative</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/P-derivation">p-derivation</a>

%F a(n) = (A006094(n))''.

%F a(n) = A068346(A006094(n))= A003415(A003415(A006094(n))).

%e (2*3)’ = 1*3+2*1 = 5; (5)’ = 1; (2^2)’ = 2*2^1 = 2*2 = 4.

%p with(numtheory); P:=proc(q) local a,b,c,p,n;

%p for n from 1 to q do a:=ithprime(n)*ithprime(n+1);

%p b:=a*add(op(2,p)/op(1,p),p=ifactors(a)[2]);

%p c:=b*add(op(2,p)/op(1,p),p=ifactors(b)[2]);

%p print(c); od; end: P(10^3); # _Paolo P. Lava_, Apr 01 2014

%o (Haskell)

%o a240052 = a068346 . a006094  -- _Reinhard Zumkeller_, Apr 15 2014

%Y Cf. A006094, A001043.

%Y Cf. A003415 (1st derivative), A068346(2nd derivative).

%K nonn

%O 1,2

%A _Freimut Marschner_, Mar 31 2014