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

5, 31, 2927, 40361, 201015517717077830328949, 13585328068403621603022853, 5692733621468679832887230172131, 3215488142498485484492183158345029261034221047849345857469577412562094716564064084247

Offset: 1

Author

Freimut Marschner, Jul 02 2014

Keywords

• nonn

Comments

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.

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n a(n) = (prime(i)#)’ i (a(n))'

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1 (3#)’ 2 1

2 (5#)’ 3 1

3 (11#)’ 5 1

4 (13#)’ 6 1

5 (61#)’ 18 1

6 (67#)’ 19 1

7 (79#)’ 22 1

8 (211#)’ 47 1

9 (269#)’ 57 1

10 (271#)’ 58 1

11 (307#)’ 63 1

12 (349#)’ 70 1

13 (367#)’ 73 1

A-number links for A109628 and A369651 added by Antti Karttunen, Feb 08 2024

Examples

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

Links

• Freimut Marschner, Table of n, a(n) for n = 1..13

Crossrefs

Cf. A000040, A002110, A024451, A003415, A109628, A244621, A369651.

Cf. also A351088.

Programs

Maple

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

Mathematica

f[n_] := Numerator[Accumulate[Table[1/Prime[i], {i, 1, n}]]];
Select[f[50], PrimeQ] (* Ivan N. Ianakiev, Jul 08 2019 *)

PARI

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

Formula

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

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

a(n) = A024451(A109628(n)). - Antti Karttunen, Feb 08 2024