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A244622 Primes in the sequence of first arithmetic derivative of primorials. 2
5, 31, 2927, 40361, 201015517717077830328949, 13585328068403621603022853, 5692733621468679832887230172131, 3215488142498485484492183158345029261034221047849345857469577412562094716564064084247 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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)#)'' 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.

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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

LINKS

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

FORMULA

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

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

EXAMPLE

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

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 *)

PROG

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

CROSSREFS

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

Sequence in context: A225158 A299887 A088548 * A278574 A062631 A156027

Adjacent sequences:  A244619 A244620 A244621 * A244623 A244624 A244625

KEYWORD

nonn

AUTHOR

Freimut Marschner, Jul 02 2014

STATUS

approved

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Last modified September 16 06:21 EDT 2019. Contains 327090 sequences. (Running on oeis4.)