cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A070826 One half of product of first n primes A000040.

Original entry on oeis.org

1, 3, 15, 105, 1155, 15015, 255255, 4849845, 111546435, 3234846615, 100280245065, 3710369067405, 152125131763605, 6541380665835015, 307444891294245705, 16294579238595022365, 961380175077106319535, 58644190679703485491635, 3929160775540133527939545, 278970415063349480483707695
Offset: 1

Views

Author

Wolfdieter Lang, May 10 2002

Keywords

Comments

Also, with offset 0, product of first n odd primes. - N. J. A. Sloane, Feb 26 2017
Identical to A002110(n)/2, n>=1.
a(n+1) is the least odd number with exactly n distinct prime divisors. - Labos Elemer, Mar 24 2003
Also, odd numbers n for which sigma(n)*phi(n)/n^2 reaches a new record low, monotonically decreasing to the lower bound 8/Pi^2. - M. F. Hasler, Jul 08 2025

Links

Crossrefs

Cf. A003266 (for Fibonacci), A070825 (for Lucas), A003046 (for Catalan).
Cf. also A002110, A024451, A060389, A091852, A276086, A203008 [= A003415(a(1+n))].
Range of A196529.

Programs

  • Maple
    a:=n->mul(ithprime(j), j=2..n):seq(a(n), n=1..17); # Zerinvary Lajos, Aug 24 2008
  • Mathematica
    Rest[ FoldList[ Times, 1, Prime[ Range[ 18]] ]]/2 (* Robert G. Wilson v, Feb 17 2004 *)
FoldList[Times, 1, Prime[Range[2, 18]]] (* Zak Seidov, Jan 26 2009 *)
  • PARI
    a(n) = prod(k=2, n, prime(k)) \\ Michel Marcus, Mar 25 2017, simplified by M. F. Hasler, Jul 09 2025
  • Python
    from sympy import primorial
def A070826(n): return primorial(n)>>1 # Chai Wah Wu, Jul 21 2022

Formula

a(n) = A002110(n)/2.
From Antti Karttunen, Feb 06 2024: (Start)
a(1) = 1, and for n > 1, a(n) = A276086(A060389(n-1)).
a(n) = A024451(n) - 2*A203008(n-1).
(End)
a(n) = A000040(n)*a(n-1) for n > 1, a(1) = 1. - M. F. Hasler, Jul 09 2025

Extensions

Formula corrected by Gary Detlefs, Dec 07 2011

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

Original entry on oeis.org

0, 1, 5, 31, 247, 2927, 40361, 716167, 14117683, 334406399, 9920878441, 314016924901, 11819186711467, 492007393304957, 21460568175640361, 1021729465586766997, 54766551458687142251, 3263815694539731437539, 201015517717077830328949, 13585328068403621603022853
Offset: 0

Views

Author

N. J. A. Sloane, Clark Kimberling

Keywords

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
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

Examples

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

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

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.
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.

Programs

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

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, 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)

Extensions

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

A060389 a(1)=p_1, a(2)=p_1 + p_1*p_2, a(3)=p_1 + p_1*p_2 + p_1*p_2*p_3, ... where p_i is the i-th prime.

Original entry on oeis.org

2, 8, 38, 248, 2558, 32588, 543098, 10242788, 233335658, 6703028888, 207263519018, 7628001653828, 311878265181038, 13394639596851068, 628284422185342478, 33217442899375387208, 1955977793053588026278
Offset: 1

Views

Author

Jason Earls, Apr 04 2001

Keywords

Comments

Partial sums of the primorials A002110 starting from 2. - Charles R Greathouse IV, Feb 05 2014
All terms are even. From a(98) on, all terms are multiples of 523. - Charles R Greathouse IV, Feb 05 2014
The only values of n for which a(n)/2 is prime are: 3, 5, 7, 11, 15, 47, 49. The corresponding 7 primes are: 19, 1279, 271549, 103631759509, 314142211092671239, 826811434211869939736645732264127163964562391958563586838409421490271014424018927729, 41839936239750050346953677118447851613901200239299781782205558511980130628486398081201749. - Amiram Eldar, May 04 2017

Examples

			a(4) = 248 because p_1 + p_1*p_2 + p_1*p_2*p_3 + p_1*p_2*p_3*p_4 = 2 + 6 + 30 + 210 = 248.
a(5) = 2558: A002110(3) = 30, A286624(4) = 85, a(2) = 8; 30*85 + 8 = 2558. - _Bob Selcoe_, Oct 12 2017

Links

Crossrefs

Cf. A002110, A070826, A084737, A143293, A203008 [= A327860(a(n))], A276085, A286624, A373158.

Programs

  • Maple
    for n from 1 to 30 do printf(`%d,`,sum(product(ithprime(i), i=1..j), j=1..n)) od:
  • Mathematica
    Accumulate[Denominator[Accumulate[1/Prime[Range[20]]]]] (* Alonso del Arte, Mar 21 2013 *)
Accumulate@ FoldList[Times, Prime@ Range@ 17] (* Michael De Vlieger, May 04 2017 *)
  • PARI
    a(n)=my(s,P=1); forprime(p=2,prime(n),s+=P*=p); s \\ Charles R Greathouse IV, Feb 05 2014

Formula

a(n) = A002110(n-2)*A286624(n-1) + a(n-3), n >= 4. - Bob Selcoe, Oct 12 2017
a(n) = A276085(A070826(1+n)) = A084737(2+n)-2 = A373158(A002110(n)). - Antti Karttunen, Feb 06 2024, Oct 28 2024

Extensions

More terms from James Sellers, Apr 05 2001
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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