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

0,3

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; 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

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.

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 and Feb 08 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.]

(End)

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-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 *)

Prog

(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
(Python)
from math import prod
from sympy import prime
def A024551(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

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.

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

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).

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

Sequence in context: A261498 A368320 A276312 * A046852 A361408 A056541

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