A244621 -id:A244621 - cp's OEIS frontend

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

N. J. A. Sloane, Clark Kimberling

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

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

S. R. Finch, Mathematical Constants, Cambridge, 2003, Sect. 2.2.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Sect. VII.28.

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

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.

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

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

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 *)

a(n) = numerator(sum(i=1, n, 1/prime(i))); \\ Michel Marcus, Sep 18 2018

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

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

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)

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

A244622 Primes in the sequence of first arithmetic derivative of primorials.

Original entry on oeis.org

5, 31, 2927, 40361, 201015517717077830328949, 13585328068403621603022853, 5692733621468679832887230172131, 3215488142498485484492183158345029261034221047849345857469577412562094716564064084247

Offset: 1

Freimut Marschner, Jul 02 2014

nonn

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.
------------------------------------------------------
n              a(n) = (prime(i)#)’   i         (a(n))'
------------------------------------------------------
            (3#)’                 2            1
            (5#)’                 3            1
           (11#)’                 5            1
           (13#)’                 6            1
           (61#)’                18            1
           (67#)’                19            1
           (79#)’                22            1
          (211#)’                47            1
          (269#)’                57            1
         (271#)’                58            1
         (307#)’                63            1
         (349#)’                70            1
         (367#)’                73            1
A-number links for A109628 and A369651 added by Antti Karttunen, Feb 08 2024

			a(1) = (3#)' = (2*3 = 6)' = 2+3 = 5.

Freimut Marschner, Table of n, a(n) for n = 1..13

Cf. A000040, A002110, A024451, A003415, A109628, A244621, A369651.

Cf. also A351088.

a(1) = (prime(2)#)' = (3#)' = (6)' = 5, (5)' = 1 ; a(4) = (prime(6)#)' = (13#)' =(30030)' = 40361, (40361)' = 1.

f[n_] := Numerator[Accumulate[Table[1/Prime[i], {i, 1, n}]]];
Select[f[50], PrimeQ] (* Ivan N. Ianakiev, Jul 08 2019 *)

lista() = {vadp = readvec("/gp/bfiles/b024451.txt"); for (i=1, #vadp, if (isprime(vadp[i]), print1(vadp[i], ", ");););} \\ Michel Marcus, Jul 05 2014

a(n) = (prime(i)#)' if (prime(i)#)'' = 1.
a(n) = (prime(i)#)' if A003415(A002110(i)) is prime or A003415(A024451(i)) = 1.
a(n) = A024451(A109628(n)). - Antti Karttunen, Feb 08 2024

cp's OEIS Frontend

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

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