A240052 2nd arithmetic derivative of products of 2 successive prime numbers (A006094).

1, 12, 16, 21, 44, 31, 60, 41, 56, 92, 72, 71, 124, 123, 140, 240, 244, 448, 121, 384, 236, 297, 176, 161, 249, 284, 247, 540, 191, 608, 221, 272, 380, 912, 520, 380, 1024, 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

Offset: 1

Author

Freimut Marschner, Mar 31 2014

Keywords

• nonn

Comments

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)’’.

Examples

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

Links

• Freimut Marschner, Table of n, a(n) for n = 1..429
• Wikipedia, Arithmetic derivative
• Wikipedia, p-derivation

Crossrefs

Cf. A006094, A001043.

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

Programs

Haskell

a240052 = a068346 . a006094  -- Reinhard Zumkeller, Apr 15 2014

Maple

with(numtheory); P:=proc(q) local a,b,c,p,n;
for n from 1 to q do a:=ithprime(n)*ithprime(n+1);
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]);
print(c); od; end: P(10^3); # Paolo P. Lava, Apr 01 2014

Formula

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

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