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

0, 32, 80, 7, 112, 0, 96, 0, 96, 272, 220, 0, 448, 48, 192, 1552, 380, 5056, 13, 4480, 608, 756, 752, 31, 45, 912, 80, 2484, 0, 3120, 31, 1312, 572, 7744, 1448, 572, 26624, 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

Offset

1,2

Comments

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.

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(A068346(n)) = A003415(A003415(A003415(A003415(n)))).

Example

(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.

Maple

with(numtheory); P:= proc(q) local a, b, c, d, f, n, p;  a:=ithprime(n)*ithprime(n+1);
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]);
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]);
print(d); od; end: P(10^4); # Paolo P. Lava, Apr 07 2014

Crossrefs

Cf. A006094, A001043, A068346, A003415, A240052, A240053.

Sequence in context: A224220 A007797 A222546 * A135269 A234140 A362044

Adjacent sequences: A240051 A240052 A240053 * A240055 A240056 A240057

Keyword

nonn

Author

Freimut Marschner, Mar 31 2014

Status

approved