A369972 -id:A369972 - 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

A369970 Numbers k such that A003415(k) is a multiple of A276086(k), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 6, 2315, 510510

Offset: 1

Antti Karttunen, Feb 07 2024

For the general dynamics of this phenomenon, see the scatter plots of A351231 and A351233.
Question: Are the terms by necessity all squarefree?
As a subsequence this sequence includes all primorials with indices k such that A024451(k) is a multiple of A000040(1+k). See A369972 and A369973.
872415232 < a(6) <= 13082761331670030 [= A369973(4)].

			2315 is included as A003415(2315) = 5+463 = 468 = 2^2 * 3^2 * 13 (note that 2315 is a semiprime = 5*463, thus its arithmetic derivative is the sum of its two prime factors), and because that 468 is a multiple of A276086(2315) = 234 = 2 * 3^2 * 13 [the exponents of primes are here read from the primorial base expansion of 2315, A049345(2315) = 100021].
510510 is included because A003415(510510) = 19*37693, which is a multiple of A276086(510510) = 19.

Index entries for sequences related to primorial base

Cf. A000040, A003415, A024451, A276086, A369972, A369973 (subsequence).
Positions of 1's in A351231, positions of 0's in A351233 and in A369971.
After the two initial terms, a subsequence of A351228.
Cf. also A358221.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA369970(n) = !(A003415(n)%A276086(n));

A369973 Primorials whose arithmetic derivative is divisible by the next larger prime not present in that primorial.

Original entry on oeis.org

1, 6, 510510, 13082761331670030, 40729680599249024150621323470, 2566376117594999414479597815340071648394470

Offset: 1

Antti Karttunen, Feb 07 2024

nonn

Primorials A002110(k) such that A003415(A002110(k)) [= A024451(k)] is a multiple of A000040(1+k).

a(7) = A002110(261202), which is too large to include here, or even in a b-file.

			The zeroth primorial, 1 = A002110(0), is included, because its arithmetic derivative 1' = A024451(0) = 0 is divisible by the next larger prime not present in the primorial, in this case by prime(1) = 2.
The primorial 510510 = prime(7)# is included, because its arithmetic derivative 510510' = A024451(7) = 716167 = 19*37693 is divisible by the next larger prime not present in the primorial, in this case by prime(8) = 19.

Index entries for sequences related to primorial numbers

Cf. A000040, A002110, A003415, A024451, A293457 (the corresponding primes), A369972.

Subsequence of A369970.

A002110(n) = prod(i=1,n,prime(i));
A024451(n) = numerator(sum(i=1, n, 1/prime(i)));
isA369972(n) = !(A024451(n)%prime(1+n));
for(n=0,2^10,if(isA369972(n),print1(A002110(n),", ")));

a(n) = A002110(A369972(n)).

A293457 Primes that divide the numerator of the sum of the reciprocals of all smaller primes.

Original entry on oeis.org

2, 5, 19, 47, 79, 109, 3667387

Offset: 1

Logan J. Kleinwaks, Oct 09 2017

Exhaustive search finds no more terms among the first 10^7 primes.

Primes p that divide A024451(A000720(p)-1). - Antti Karttunen, Feb 08 2024

			Since 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 = 716167/510510 and 19 divides 716167, 19 is in the sequence.
Since there are no primes less than 2, the sum of their reciprocals is 0/1, and as 2 divides 0, it is therefore included as the first term of this sequence. - _Antti Karttunen_, Feb 08 2024

Cf. A000040, A000720, A024451, A120289, A369972, A369973.

lista(nn) = my(s = 0); forprime(p=2, nn, if (!(numerator(s) % p), print1(p, ", ")); s += 1/p; ); \\ Michel Marcus, Oct 09 2017, edited for the new, more inclusive definition by Antti Karttunen, Feb 08 2024

a(n) = A000040(1+A369972(n)). - Antti Karttunen, Feb 08 2024

Relaxed the definition to include 2 as the first term - Antti Karttunen, Feb 08 2024

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

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A369970 Numbers k such that A003415(k) is a multiple of A276086(k), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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A369973 Primorials whose arithmetic derivative is divisible by the next larger prime not present in that primorial.

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A293457 Primes that divide the numerator of the sum of the reciprocals of all smaller primes.

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