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

1,2

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)’’.

Links

Freimut Marschner, Table of n, a(n) for n = 1..429

Wikipedia, Arithmetic derivative

Wikipedia, p-derivation

Formula

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

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

Example

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

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

Prog

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

Crossrefs

Cf. A006094, A001043.

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

Sequence in context: A243538 A183052 A344079 * A363798 A070329 A064695

Adjacent sequences: A240049 A240050 A240051 * A240053 A240054 A240055

Keyword

nonn

Author

Freimut Marschner, Mar 31 2014

Status

approved