A258566 -id:A258566 - 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

A077011 Triangle read by rows: T(n,k) = A002110(n)/prime(n+1-k), k = 1..n.

Original entry on oeis.org

1, 2, 3, 6, 10, 15, 30, 42, 70, 105, 210, 330, 462, 770, 1155, 2310, 2730, 4290, 6006, 10010, 15015, 30030, 39270, 46410, 72930, 102102, 170170, 255255, 510510, 570570, 746130, 881790, 1385670, 1939938, 3233230, 4849845, 9699690, 11741730

Offset: 1

Amarnath Murthy, Oct 26 2002

Original Name was: "Triangle in which the n-th row contains all possible products of n-1 of the first n primes in ascending order."
A024451(n) gives the sum of the n-th row.
When the triangle is parsed in blocks of ascending length, as shown in the example, there is the following interpretation: The integers Z regarded as a module over themselves contain unshortenable generating sets of different lengths, in fact, infinitely many of each desired length. Each of the blocks is the minimal example of an unshortenable generating set of the respective length. For example, {6,10,15} generates Z as 1=6+10-15. However, removing one of the numbers leaves two numbers that are not relatively prime, precluding generation of Z. An analogous argument succeeds for all other blocks alike. Each block contains numbers such that there is no prime factor common to all. Taking differences sufficiently often one ends up with two coprime numbers whence the generating property follows from Bézout's theorem. If just one number is removed from the set, relative primality is lost. The minimality of the numbers used in each block is evident from the construction. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 04 2006

			Triangle begins:
      1;
      2,     3;
      6,    10,    15;
     30,    42,    70,   105;
    210,   330,   462,   770,   1155;
   2310,  2730,  4290,  6006,  10010,  15015;
  30030, 39270, 46410, 72930, 102102, 170170, 255255;
  ...

Alois P. Heinz, Rows n = 1..130, flattened

Row sums give A024451.
Reversal of A258566.
Cf. A002110, A089633, A372666.

T:= proc(n) local t;
      t:= mul(ithprime(i), i=1..n);
      seq(t/ithprime(n-i), i=0..n-1)
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Jun 04 2012

T[n_] := Module[{t = Product[Prime[i], {i, 1, n}]}, Table[t/Prime[n - i], {i, 0, n - 1}]];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, May 19 2016, translated from Maple *)

T(n,k) = vecprod(primes(n))/prime(n+1-k); \\ Michel Marcus, May 19 2024

A089633(n-1) = Sum_{p | n} 2^(pi(p) - 1) for n > 1, pi(x) = A000720(x). - Michael De Vlieger, May 19 2024

More terms from Sascha Kurz, Jan 26 2003

Name changed by David James Sycamore, May 19 2024

A286947 Triangle read by rows in which row(n) = {T(n, k)} is the lexicographically earliest list of n numbers such that adding 1 to some T(n, k) gives a row of numbers each divisible by prime(k).

Original entry on oeis.org

1, 3, 2, 15, 20, 24, 105, 140, 84, 90, 1155, 770, 924, 1980, 2100, 15015, 10010, 24024, 4290, 13650, 23100, 255255, 340340, 204204, 364650, 464100, 353430, 60060, 4849845, 6466460, 5819814, 1385670, 3527160, 5969040, 570570, 510510, 111546435, 74364290, 44618574, 127481640, 81124680, 102965940, 39369330, 58708650, 29099070

Offset: 1

David A. Corneth, May 17 2017

1 + the Rowsum of row(n) gives a multiple of A002110(n).
c = Product_{i=1..n} prime(i)^T(n, i) is the least term such that prime(i) * c is a prime(i)-th power. First such terms are 2, 72, 6810125783203125000000000000000, ... which relates this sequence to A286930.
T(n,k) is a multiple of A258566(n,k). - Peter Munn, Jan 13 2018

			Row(1): [1]
Row(2): [3, 2]
Row(3): [15, 20, 24]
Row(4): [105, 140, 84, 90]
Row(5): [1155, 770, 924, 1980, 2100]
Row(6): [15015, 10010, 24024, 4290, 13650, 23100]
Row(7): [255255, 340340, 204204, 364650, 464100, 353430, 60060]
Row(8): [4849845, 6466460, 5819814, 1385670, 3527160, 5969040, 570570, 510510]
Row(4) = [105, 140, 84, 90].
Adding 1 to T(4, 1) gives [106,140,84,90], all elements divisible by prime(1) = 2.
Adding 1 to T(4, 2) gives [105,141,84,90], all elements divisible by prime(2) = 3.
Adding 1 to T(4, 3) gives [105,140,85,90], all elements divisible by prime(3) = 5.
Adding 1 to T(4, 4) gives [105,140,84,91], all elements divisible by prime(4) = 7.
The sum of elements in row 3 is 15 + 20 + 24 = 59. 59 + 1 = 60, a multiple of A002110(3) = 30.

Cf. A002110, A075306, A258566, A286930.

row(n) = my(pr=primes(n), p = prod(i=1, #pr, pr[i]), res=vector(n, i, lift(chinese(Mod(-1, pr[i]), Mod(0, p/pr[i]))))); res

T(n, 1) = A002110(n) / 2.

For n >= 2, T(n,n) = A075306(n-1) - 1. - Peter Munn, Jan 13 2018

Name corrected by Peter Munn, Jan 12 2018

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

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A077011 Triangle read by rows: T(n,k) = A002110(n)/prime(n+1-k), k = 1..n.

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A286947 Triangle read by rows in which row(n) = {T(n, k)} is the lexicographically earliest list of n numbers such that adding 1 to some T(n, k) gives a row of numbers each divisible by prime(k).

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