A240053 -id:A240053 - cp's OEIS frontend

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

Freimut Marschner, Mar 31 2014

nonn

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.

			(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.

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

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

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

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

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula