A024451 a(n) is the numerator of Sum_{i = 1..n} 1/prime(i).

0, 1, 5, 31, 247, 2927, 40361, 716167, 14117683, 334406399, 9920878441, 314016924901, 11819186711467, 492007393304957, 21460568175640361, 1021729465586766997, 54766551458687142251, 3263815694539731437539, 201015517717077830328949, 13585328068403621603022853

Offset: 0

Author

N. J. A. Sloane, Clark Kimberling

Keywords

• nonn
• frac
• easy
• nice

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; A250130 gives the numerators. - 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

The above conjecture would imply that for n > 0, gcd(a(n), A369651(n)) = 1. See corollary 2 on the page 4 of Ufnarovski-Åhlander paper. - Antti Karttunen, Jan 31 2024

Apart from the initial 0, a subsequence of A048103. Proof: For all primes p, when i >= A000720(p), neither p itself nor p^p divides a(i) [implied by Henry Bottomley's Sep 27 2006 formula], but neither does p^p divide a(i) when 0 < i < A000720(p), as then p^p > a(i). See A074107, which gives an upper bound for this sequence. - Antti Karttunen, Nov 19 2024

Examples

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

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

• Alois P. Heinz, Table of n, a(n) for n = 0..350 (terms n = 1..100 from T. D. Noe)
• Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.
• Index entries for sequences related to primorial numbers

Crossrefs

Denominators are A002110.

See also A106830/A034386, A241189/A241190, A241191/A241192, A061015/A061742, A115963/A115964, A250133/A296358, and A096795/A051451, A354417/A354418, A354859/A354860.

Row sums of A077011 and A258566.

Subsequence of A048103 (after the initial 0).

Cf. A003415, A002110, A070826, A143293, A203008, A250130, A327860, A348301, A369651.

Cf. A053144 (a lower bound), A074107 (an upper bound).

Cf. A109628 (indices k where a(k) is prime), A244622 (corresponding primes), A244621 (a(n) mod 12).

Cf. A369972 (k where prime(1+k)|a(k)), A369973 (corresponding primorials), A293457 (corresponding primes), A377992 (antiderivatives of the terms > 1 of this sequence).

Cf. also A223037, A260615, A274070, A327978, A353299, A353534, A356253.

Programs

Magma

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

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-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 *)

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

Python

from math import prod
from sympy import prime
def A024451(n):
    q = prod(plist:=tuple(prime(i) for i in range(1,n+1)))
    return sum(q//p for p in plist) # Chai Wah Wu, Nov 03 2022

Formula

Limit_{n->oo} (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

From Antti Karttunen, Jan 31 2024, Feb 08 2024 and Nov 19 2024: (Start)

a(0) = 0, for n > 0, a(n) = 2*A203008(n-1) + A070826(n).

For n > 0, a(n) = A327860(A143293(n-1)).

For n > 0, a(n) = A348301(n) + A002110(n).

For n = 3..175, a(n) = A356253(A002110(n)). [See comments in A356253.]

For n >= 0, A053144(n) <= a(n) <= A074107(n) < A070826(1+n).

(End)

Extensions

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