A240052 -id:A240052 - cp's OEIS frontend

A240053 3rd Arithmetic derivation of products of 2 successive prime numbers (A006094).

0, 16, 32, 10, 48, 1, 92, 1, 92, 96, 156, 1, 128, 44, 188, 608, 248, 1408, 22, 1472, 240, 324, 368, 30, 86, 288, 32, 1188, 1, 1552, 30, 560, 476, 2176, 924, 476, 5120, 60, 432, 2176, 1148, 512, 4480, 1, 1300, 324, 1, 391, 1052, 46, 720, 3232, 672, 2304, 1448, 860, 2484, 1036, 226, 768, 7232, 1628

Offset: 1

Freimut Marschner, Mar 31 2014

nonn

The first arithmetic derivation of products of 2 successive prime numbers (A006094) is the sum of 2 successive prime numbers (A001043). A001043 = (A006094)’. The second arithmetic derivation is (A240052) = (A001043)’ = (A006094)’’. The third arithmetic derivation of products of 2 successive prime numbers (A006094) is a(n) = (A240052)’ = (A001043)’’ = (A006094)’’’.

			a(12)=(A006094(12))'''=(37*41)'''=(A001043(12))''=(78)''=(71)'=1;
a(14)=(A006094(14))'''=(43*47)'''=(A001043(12))''=(90)''=(123)'=44.

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

Cf. A006094, A001043, A240052.

Cf. A003415 (1st derivative), A068346 (2nd derivative), A099306 (3rd derivative).

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

a(n) = (A006094(n))’’’.
a(n) = A099306(A006094(n)).
a(n) = A003415(A240052(n)).

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

Original entry on oeis.org

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

A240053 3rd Arithmetic derivation of products of 2 successive prime numbers (A006094).

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