A109628 -id:A109628 - 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

A351088 Numbers k such that A327860(k) is reachable from k by iterating the arithmetic derivative (A003415) and there are no terms with p^p-factors on the path there.

Original entry on oeis.org

0, 1, 2, 6, 7, 8, 30, 2310, 2556, 30030, 223092870

Offset: 1

Antti Karttunen, Feb 05 2022

Sequence includes also the terms for which no iterations are needed (when k is already equal to A327860(k)), thus A328110 is a subsequence. The other terms (and also 1) seem to be the intersection of primorials (A002110) with sequence A099308. This includes terms A002110(A109628(n)), whose arithmetic derivatives are in A244622.
The numbers k for which A276086(k) is reachable from k by iterating A003415 form a subsequence of this sequence, but so far only one term is known: 6, for which A276086(6) = A003415(6) = 5. (See A351228). It would be interesting to know whether there are more such terms, especially terms that require more than one iteration of A003415.
Question: The eleven known terms are all sums of distinct primorials (in A276156), i.e., contain only digits 0's and 1's in primorial base. Is this a necessary property for the terms of this sequence (and also for A328110)? - Antti Karttunen, Feb 04 2024, corrected May 11 2024.

Cf. A002110, A003415, A099308, A109628, A244622, A276086, A276156, A328110 (subsequence), A327860.

Cf. also A327969, A351089, A351228, A358215.

A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s)); \\ Like A003415, but return zero also for n that have p^p-factor(s).
A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
\\ This simple program doesn't check for any hypothetical p^p-free A003415-loops (they are so rare that they are conjectured not to exist at all):
isA351088(n) = if(!n, 1, my(g=A327860(n)); while(n>0, if(n==g, return(1)); n = A003415checked(n)); (n));

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

A369651 The second arithmetic derivative of primorial numbers.

Original entry on oeis.org

0, 0, 1, 1, 32, 1, 1, 37712, 381596, 9835475, 36880858, 5221912091, 149759057759, 186024798, 456607833524310, 655374897748042, 32188411533060, 3002590335363139484, 1, 1, 1489192548103758778136571, 905624730150525686386253700, 1, 110832638787270947169636975150, 331748330219014906068858, 349669183443106226663011852002384

Offset: 0

Antti Karttunen, Jan 31 2024

nonn

See comments in A024451.

Antti Karttunen, Table of n, a(n) for n = 0..49

Cf. A002110, A003415, A024451, A068346, A109628 (positions of 1's).

Cf. also A368702.

A002110(n) = prod(i=1,n,prime(i));
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A369651(n) = A003415(A003415(A002110(n)));

a(n) = A003415(A024451(n)) = A068346(A002110(n)).

A369972 Numbers k such that (prime(k)#)' is a multiple of prime(1+k), where prime(k)# means the k-th primorial, A002110(k), and ' stands for taking the arithmetic derivative, A003415.

Original entry on oeis.org

0, 2, 7, 14, 21, 28, 261202

Offset: 1

Antti Karttunen, Feb 07 2024

Numbers k for which A024451(k) is a multiple of A000040(1+k).

			7 is included because the primorial prime(7)# = A002110(7) = 510510 has as its arithmetic derivative 510510' = A024451(7) = 716167 = 19*37693, which 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, A000720, A024451, A293457 (corresponding primes), A369970, A369973 (corresponding primorials).

Cf. also A109628.

A024451(n) = numerator(sum(i=1, n, 1/prime(i)));
isA369972(n) = !(A024451(n)%prime(1+n));

a(n) = A000720(A293457(n)) - 1.

Found a(7) by computing it as A000720(A293457(7))-1. - 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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A351088 Numbers k such that A327860(k) is reachable from k by iterating the arithmetic derivative (A003415) and there are no terms with p^p-factors on the path there.

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

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A369651 The second arithmetic derivative of primorial numbers.

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A369972 Numbers k such that (prime(k)#)' is a multiple of prime(1+k), where prime(k)# means the k-th primorial, A002110(k), and ' stands for taking the arithmetic derivative, A003415.

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A125707 Numbers k such that A024528(k) is prime.

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A125708 Numbers k such that A115963(k) is prime.

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