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A024451
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a(n) is the numerator of Sum_{i = 1..n} 1/prime(i).
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46
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0, 1, 5, 31, 247, 2927, 40361, 716167, 14117683, 334406399, 9920878441, 314016924901, 11819186711467, 492007393304957, 21460568175640361, 1021729465586766997, 54766551458687142251, 3263815694539731437539, 201015517717077830328949, 13585328068403621603022853
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OFFSET
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0,3
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COMMENTS
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Arithmetic derivative of p#: a(n) = A003415(A002110(n)). - Reinhard Zumkeller, Feb 25 2002
(n-1)-st elementary symmetric functions of first n primes; see Mathematica section. - Clark Kimberling, Dec 29 2011
Denominators of the harmonic mean of the first n primes. - Colin Barker, Nov 14 2014
Let Pn(n) = A002110 denote the primorial function. The average number of distinct prime factors <= prime(n) in the natural numbers up to Pn(n) is equal to Sum_{i = 1..n} 1/prime(i). - Jamie Morken, Sep 17 2018
Conjecture: All terms are squarefree numbers. - Nicolas Bělohoubek, Apr 13 2022
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, Sect. 2.2.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Sect. VII.28.
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LINKS
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T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..350 (terms n = 1..100 from T. D. Noe)
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FORMULA
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lim_{n -> infinity} (Sum_{p <= n} 1/p - log log n) = 0.2614972... = A077761.
a(n) = (Product_{i=1..n} prime(i))*(Sum_{i=1..n} 1/prime(i)). - Benoit Cloitre, Jan 30 2002
(n+1)-st elementary symmetric function of the first n primes.
a(n) = a(n-1)*A000040(n) + A002110(n-1). - Henry Bottomley, Sep 27 2006
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EXAMPLE
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0/1, 1/2, 5/6, 31/30, 247/210, 2927/2310, 40361/30030, 716167/510510, 14117683/9699690, ...
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MAPLE
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h:= n-> add(1/(ithprime(i)), i=1..n);
t1:=[seq(h(n), n=0..50)];
t1a:=map(numer, t1); # A024451
t1b:=map(denom, t1); # A002110 - N. J. A. Sloane, Apr 25 2014
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MATHEMATICA
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a[n_] := Numerator @ Sum[1/Prime[i], {i, n}]; Array[a, 18] (* Jean-François Alcover, Apr 11 2011 *)
f[k_] := Prime[k]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A024451 *)
(* Clark Kimberling, Dec 29 2011 *)
Numerator[Accumulate[1/Prime[Range[20]]]] (* Harvey P. Dale, Apr 11 2012 *)
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PROG
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(MAGMA) [ Numerator(&+[ NthPrime(k)^-1: k in [1..n]]): n in [1..18] ]; // Bruno Berselli, Apr 11 2011
(PARI) a(n) = numerator(sum(i=1, n, 1/prime(i))); \\ Michel Marcus, Sep 18 2018
(Python)
from sympy import prime
from fractions import Fraction
def a(n): return sum(Fraction(1, prime(k)) for k in range(1, n+1)).numerator
print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 12 2021
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CROSSREFS
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Denominators are A002110. See also A106830/A034386, A241189/A241190, A241191/A241192, A061015/A061742, A115963/A115964, A250133/A296358.
Sequence in context: A294214 A261498 A276312 * A046852 A056541 A291885
Adjacent sequences: A024448 A024449 A024450 * A024452 A024453 A024454
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KEYWORD
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nonn,frac,easy,nice
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AUTHOR
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N. J. A. Sloane, Clark Kimberling
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EXTENSIONS
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a(0)=0 prepended by Alois P. Heinz, Jun 26 2015
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STATUS
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approved
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