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A244622 Primes in the sequence of first arithmetic derivative of primorials. 2

%I

%S 5,31,2927,40361,201015517717077830328949,13585328068403621603022853,

%T 5692733621468679832887230172131,

%U 3215488142498485484492183158345029261034221047849345857469577412562094716564064084247

%N Primes in the sequence of first arithmetic derivative of primorials.

%C A002110 is the sequence of primorial numbers (product of consecutive prime numbers, written prime(n)#). A024451 = numerator of Sum_{i = 1..n} 1/prime(i) is the first arithmetic derivative of prime(n)#, written (prime(n)#)'. The second arithmetic derivative of prime(n)#, written (prime(n)#)'' is 1 if (prime(n)#)' is prime. This case leads to a selection of 13 primorials out of the first 100 primorials. The table shows the counting number n of this selection, the primorial notation, the index i used in A002110 and A024451 and the 2nd arithmetic derivative of the 13 prime numbers of A024451. Remark: i is the prime number index of A000040.

%C ------------------------------------------------------

%C n a(n) = (prime(i)#)’ i (a(n))'

%C ------------------------------------------------------

%C 1 (3#)’ 2 1

%C 2 (5#)’ 3 1

%C 3 (11#)’ 5 1

%C 4 (13#)’ 6 1

%C 5 (61#)’ 18 1

%C 6 (67#)’ 19 1

%C 7 (79#)’ 22 1

%C 8 (211#)’ 47 1

%C 9 (269#)’ 57 1

%C 10 (271#)’ 58 1

%C 11 (307#)’ 63 1

%C 12 (349#)’ 70 1

%C 13 (367#)’ 73 1

%H Freimut Marschner, <a href="/A244622/b244622.txt">Table of n, a(n) for n = 1..13</a>

%F a(n) = (prime(i)#)' if (prime(i)#)'' = 1.

%F a(n) = (prime(i)#)' if A003415(A002110(i)) is prime or A003415(A024451(i)) = 1.

%e a(1) = (3#)' = (2*3 = 6)' = 2+3 = 5.

%p a(1) = (prime(2)#)' = (3#)' = (6)' = 5, (5)' = 1 ; a(4) = (prime(6)#)' = (13#)' =(30030)' = 40361, (40361)' = 1.

%t f[n_] := Numerator[Accumulate[Table[1/Prime[i], {i, 1, n}]]];

%t Select[f[50], PrimeQ] (* _Ivan N. Ianakiev_, Jul 08 2019 *)

%o (PARI) lista() = {vadp = readvec("/gp/bfiles/b024451.txt"); for (i=1, #vadp, if (isprime(vadp[i]), print1(vadp[i], ", ");););} \\ _Michel Marcus_, Jul 05 2014

%Y Cf. A002110, A024451, A003415, A000040, A244621.

%K nonn

%O 1,1

%A _Freimut Marschner_, Jul 02 2014

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Last modified October 21 11:05 EDT 2019. Contains 328294 sequences. (Running on oeis4.)