cp's OEIS Frontend

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

%I A244622 #36 Feb 09 2024 08:43:22
%S A244622 5,31,2927,40361,201015517717077830328949,13585328068403621603022853,
%T A244622 5692733621468679832887230172131,
%U A244622 3215488142498485484492183158345029261034221047849345857469577412562094716564064084247
%N A244622 Primes in the sequence of first arithmetic derivative of primorials.
%C A244622 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)#)'' [= A369651(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 [= A109628(n)] is the prime number index of A000040.
%C A244622 ------------------------------------------------------
%C A244622 n              a(n) = (prime(i)#)’   i         (a(n))'
%C A244622 ------------------------------------------------------
%C A244622 1              (3#)’                 2            1
%C A244622 2              (5#)’                 3            1
%C A244622 3             (11#)’                 5            1
%C A244622 4             (13#)’                 6            1
%C A244622 5             (61#)’                18            1
%C A244622 6             (67#)’                19            1
%C A244622 7             (79#)’                22            1
%C A244622 8            (211#)’                47            1
%C A244622 9            (269#)’                57            1
%C A244622 10           (271#)’                58            1
%C A244622 11           (307#)’                63            1
%C A244622 12           (349#)’                70            1
%C A244622 13           (367#)’                73            1
%C A244622 A-number links for A109628 and A369651 added by _Antti Karttunen_, Feb 08 2024
%H A244622 Freimut Marschner, <a href="/A244622/b244622.txt">Table of n, a(n) for n = 1..13</a>
%F A244622 a(n) = (prime(i)#)' if (prime(i)#)'' = 1.
%F A244622 a(n) = (prime(i)#)' if A003415(A002110(i)) is prime or A003415(A024451(i)) = 1.
%F A244622 a(n) = A024451(A109628(n)). - _Antti Karttunen_, Feb 08 2024
%e A244622 a(1) = (3#)' = (2*3 = 6)' = 2+3 = 5.
%p A244622 a(1) = (prime(2)#)' = (3#)' = (6)' = 5, (5)' = 1 ; a(4) = (prime(6)#)' = (13#)' =(30030)' = 40361, (40361)' = 1.
%t A244622 f[n_] := Numerator[Accumulate[Table[1/Prime[i], {i, 1, n}]]];
%t A244622 Select[f[50], PrimeQ] (* _Ivan N. Ianakiev_, Jul 08 2019 *)
%o A244622 (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 A244622 Cf. A000040, A002110, A024451, A003415, A109628, A244621, A369651.
%Y A244622 Cf. also A351088.
%K A244622 nonn
%O A244622 1,1
%A A244622 _Freimut Marschner_, Jul 02 2014