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A024451 a(n) is the numerator of Sum_{i = 1..n} 1/prime(i). 33
0, 1, 5, 31, 247, 2927, 40361, 716167, 14117683, 334406399, 9920878441, 314016924901, 11819186711467, 492007393304957, 21460568175640361, 1021729465586766997, 54766551458687142251, 3263815694539731437539, 201015517717077830328949, 13585328068403621603022853 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, Sect. 2.2.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Sect. VII.28.

LINKS

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..350 (terms n = 1..100 from T. D. Noe)

FORMULA

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

EXAMPLE

0/1, 1/2, 5/6, 31/30, 247/210, 2927/2310, 40361/30030, 716167/510510, 14117683/9699690, ...

MAPLE

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

MATHEMATICA

a[n_] := Numerator @ Sum[1/Prime[i], {i, n}]; Array[a, 18]  (* Jean-Fran├žois 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 *)

PROG

(MAGMA) [ Numerator(&+[ NthPrime(k)^-1: k in [1..n]]): n in [1..18] ];  // Bruno Berselli, Apr 11 2011

CROSSREFS

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

KEYWORD

nonn,frac,easy,nice

AUTHOR

N. J. A. Sloane, Clark Kimberling

EXTENSIONS

a(0)=0 prepended by Alois P. Heinz, Jun 26 2015

STATUS

approved

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Last modified February 19 02:06 EST 2018. Contains 299330 sequences. (Running on oeis4.)