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A244622
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Primes in the sequence of first arithmetic derivative of primorials.
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5
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5, 31, 2927, 40361, 201015517717077830328949, 13585328068403621603022853, 5692733621468679832887230172131, 3215488142498485484492183158345029261034221047849345857469577412562094716564064084247
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) = (prime(i)#)' if (prime(i)#)'' = 1.
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EXAMPLE
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a(1) = (3#)' = (2*3 = 6)' = 2+3 = 5.
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MAPLE
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a(1) = (prime(2)#)' = (3#)' = (6)' = 5, (5)' = 1 ; a(4) = (prime(6)#)' = (13#)' =(30030)' = 40361, (40361)' = 1.
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MATHEMATICA
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f[n_] := Numerator[Accumulate[Table[1/Prime[i], {i, 1, n}]]];
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PROG
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(PARI) lista() = {vadp = readvec("/gp/bfiles/b024451.txt"); for (i=1, #vadp, if (isprime(vadp[i]), print1(vadp[i], ", "); ); ); } \\ Michel Marcus, Jul 05 2014
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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