A061015 -id:A061015 - cp's OEIS frontend

A092062 Numbers k such that A061015(k) is prime.

2, 10, 18, 36, 90, 759

Offset: 1

Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Feb 20 2004

a(6) > 447 for a(6) the numerator has more than 2673 digits.

a(7) > 1850. - Michael S. Branicky, Jun 27 2022

			1/2^2 = 1/4 but 1 is not prime, 1/2^2 + 1/3^2 = 13/36 and 13 is prime so a(1)=2.

Cf. A061015.

sm(n)= s=0;for(i=1,n,s=s+1/(prime(i)^2));return(s);
for (i=1,400,if(isprime(numerator(sm(i))),print1(i,",")))

# uses A061015gen() and imports from A061015
from sympy import isprime
def agen():
    yield from (k for k, ak in enumerate(A061015gen(), 1) if isprime(ak))
print(list(islice(agen(), 5))) # Michael S. Branicky, Jun 27 2022

Numbers k such that numerator of (Sum_{i=1..k} 1/prime(i)^2) is prime

a(6) from Alexander Adamchuk, Sep 16 2010

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

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

A115964 Denominator of Sum_{i=1..n} 1/prime(i)^3.

Original entry on oeis.org

8, 216, 27000, 9261000, 12326391000, 27081081027000, 133049351085651000, 912585499096480209000, 11103427767506874702903000, 270801499821725167129101267000, 8067447481189014453943055845197000

Offset: 1

Jonathan Vos Post, Mar 14 2006

Numerators are in A115963.

Also the primorials cubed. - Reikku Kulon, Sep 18 2008

			1/8, 35/216, 4591/27000, 1601713/9261000, 2141141003/12326391000, 4716413174591/27081081027000.

Cf. A115963 (numerators).
Cf. A024451 (numerator of sum_{i=1..n} 1/prime(i)), A002110 (primorial, also denominator of sum_{i=1..n} 1/prime(i)), A061015 (numerator of sum_{i=1..n} 1/prime(i)^2).
Cf. A000040, A075986/A075987, A106830/A034386.
Cf. A061742, A100778. - Reikku Kulon, Sep 18 2008

a[n_]:=Product[Prime[i]^3, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008 *)

a(n) = denominator of Sum_{i=1..n} 1/A000040(i)^3.

a(n) = A002110(n)^3. - Reikku Kulon, Sep 18 2008

A075986 Numerator of 1+1/prime(1)^2+ ... + 1/prime(n)^2 where prime(k) is the k-th prime.

Original entry on oeis.org

1, 5, 49, 1261, 62689, 7629469, 1294716361, 375074829229, 135662633811769, 71859617272521901, 60483708554835755641, 58166700851687469003901, 79670437976161330893757369, 133981073592392620630139873389

Offset: 0

Zak Seidov, Sep 28 2002

The sum is similar to that in A061015 with an additional 1. The sum in the definition has limit about 1.45224742. The case of reciprocal cubes is in A075987.

For n>0 a(n) is the determinant of the n X n matrix with elements M[i,j] = 1+Prime[i]^2 if i=j and 1 otherwise. - Alexander Adamchuk, Jul 08 2006

			a(2) = 49 so a(3) = 49*p(3)^2 + (2*3)^2 = 49*25 + 36 = 1261.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 94-98.

Steven R. Finch, Meissel-Mertens Constants [Broken link]
Steven R. Finch, Meissel-Mertens Constants [From the Wayback machine]

Cf. A061015, A075987, A024528.

Table[Det[DiagonalMatrix[Table[Prime[i]^2,{i,1,n}]]+1],{n,1,15}] (* Alexander Adamchuk, Jul 08 2006 *)
Accumulate[Join[{1},1/Prime[Range[20]]^2]]//Numerator (* Harvey P. Dale, May 08 2023 *)

a(0)=1; a(n)=a(n-1)*prime(n)^2+(prime(1)*...*prime(n-1))^2.

Edited by Dean Hickerson, Sep 30 2002

A115963 Numerator of Sum_{i=1..n} 1/prime(i)^3.

Original entry on oeis.org

1, 35, 4591, 1601713, 2141141003, 4716413174591, 23198819007792583, 159253748925534977797, 1938552948676080555065099, 47290471293028435532185602511, 1409101231790431848106470385672201

Offset: 1

Jonathan Vos Post, Mar 14 2006

Denominators = A115964. See also: A024451 Numerator of Sum_{i=1..n} 1/prime(i). A002110 Primorial [denominator of Numerator of Sum_{i=1..n} 1/prime(i)]. A061015 Numerator of Sum_{i=1..n} 1/prime(i)^2.

			1/8, 35/216, 4591/27000, 1601713/9261000, 2141141003/12326391000, 4716413174591/27081081027000.

Cf. A000040, A075986/A075987, A106830/A034386.

Accumulate[1/Prime[Range[20]]^3]//Numerator (* Harvey P. Dale, Dec 30 2024 *)

a(n) = Numerator of Sum_{i=1..n} 1/A000040(i)^3.

A241189 Numerator of Sum_{i=1..n} 1/(prime(i)*prime(i+1)).

Original entry on oeis.org

1, 7, 11, 127, 1693, 29243, 561623, 13019431, 379503437, 11809225121, 438235268123, 18007758091069, 775817745542929, 36524284093223105, 1938403609207158571, 2160165866032831207, 131893095784520401909, 8844093116997411126541, 628373208972323386101329, 45900898298568589325230523

Offset: 1

N. J. A. Sloane, Apr 25 2014, based on a suggestion from Timothy Varghese

a(371) has 1002 decimal digits. - Michael De Vlieger, Jan 27 2016

			1/6, 7/30, 11/42, 127/462, 1693/6006, 29243/102102, 561623/1939938, 13019431/44618574, 379503437/1293938646, 11809225121/40112098026, 438235268123/1484147626962, ...

Michael De Vlieger, Table of n, a(n) for n = 1..370

Cf. A024451/A002110, A241189/A241190, A241191/A241192.

See also A061015/A061742, A115963/A115964.

g:= n-> add(1/(ithprime(i)*ithprime(i+1)),i=1..n);
t1:=[seq(g(n),n=1..20)];
t1a:=map(numer,t1); # A241189
t1b:=map(denom,t1); # A241190

Table[Numerator@ Sum[1/(Prime[i + 1] Prime@ i), {i, n}], {n, 20}] (* Michael De Vlieger, Jan 27 2016 *)
Accumulate[1/#&/@(Times@@@Partition[Prime[Range[25]],2,1])]//Numerator (* Harvey P. Dale, Mar 14 2023 *)

A075987 Numerator(1+1/prime(1)^3+ ... + 1/prime(n)^3) where prime(k) is the k-th prime.

Original entry on oeis.org

1, 9, 251, 31591, 10862713, 14467532003, 31797494201591, 156248170093443583, 1071839248022015186797, 13041980716182955257968099, 318091971114753602661286869511, 9476548712979446302049526230869201

Offset: 0

Zak Seidov, Sep 28 2002

nonn

The sum in the sequence has limit 1.1747626393. The case of reciprocal squares is in A075986.

For n>0 a(n) is the determinant of the n X n matrix with elements M[i,j] = 1+prime(i)^3 if i=j and 1 otherwise. - Alexander Adamchuk, Jul 08 2006

			a(2) = 251 so a(3) = 251*p(3)^3 + (2*3)^3 = 251*125 + 216 = 31591.

Simon Plouffe, Plouffe's Inverter

Cf. A061015, A075986.

Table[Det[DiagonalMatrix[Table[Prime[i]^3,{i,1,n}]]+1],{n,1,15}] (* Alexander Adamchuk, Jul 08 2006 *)

a(n) = numerator(1 + sum(k=1, n, 1/prime(k)^3)); \\ Michel Marcus, May 31 2022

a(0) = 1; a(n) = a(n-1)*prime(n)^3+(prime(1)*...*prime(n-1))^3.

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

Original entry on oeis.org

3, 5, 9, 43, 150, 300, 516, 1254

Offset: 1

Alexander Adamchuk, Feb 01 2007, Mar 01 2007

A115963(n) is the numerator of Sum_{k=1..n} 1/prime(k)^3.

a(9) > 5000. - Michael S. Branicky, Aug 04 2024

Cf. A115963, A024451, A092062, A061015, A109628.

f=0;Do[p=Prime[n];f=f+1/p^3;g=Numerator[f];If[PrimeQ[g],Print[{n,p,g}]],{n,1,50}]

a(6) from Alexander Adamchuk, Sep 16 2010

a(7)-a(8) from Amiram Eldar, Feb 18 2019

A126225 Least number k > 0 such that the numerator of Sum_{i=1..k} 1/prime(i)^n is a prime.

Original entry on oeis.org

2, 2, 3, 2, 3, 5, 3, 11, 3, 22

Offset: 1

Alexander Adamchuk, Mar 08 2007

a(12) > 80, a(13) = 30, a(14) = 16, a(18) = 7, a(19) = 3. - J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010
a(11) > 200, a(12) > 200. - Michel Marcus, May 27 2019
If they exist, a(11) > 1263; a(17) > 954; a(22) > 795; a(23) > 720; a(25) > 570; a(12) = 799, a(15) = 313, a(16) = 780, a(20) = 433, a(21) = 7, a(24) = 4, a(27) = 12, a(29) = 37. - J.W.L. (Jan) Eerland, Jan 26 2023

			a(1) = 2 corresponds to A024451(2) = 5, a prime.
a(2) = 2 corresponds to A061015(2) = 13, a prime.

Cf. A024451 (1/p), A061015 (1/p^2), A115963 (1/p^3).

a[n_] := Block[{i = 1, sum = 0}, While[True, sum += 1/Prime[i]^n; If[PrimeQ[Numerator@sum], Return[i]]; i++ ]] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)
Table[y[x_,y_]:=Numerator[FullSimplify[Sum[1/Prime[m]^x,{m,1,y}]]];k=1;Monitor[Parallelize[While[True,If[PrimeQ[y[n,k]],Break[]];k++];k],k],{n,1,10}] (* J.W.L. (Jan) Eerland, Jan 25 2023 *)

a(n) = {my(k=1, s=1/prime(k)^n); while (! isprime(numerator(s)), k++; s += 1/prime(k)^n); k;} \\ Michel Marcus, May 27 2019

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A092062 Numbers k such that A061015(k) is prime.

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

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A115964 Denominator of Sum_{i=1..n} 1/prime(i)^3.

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A075986 Numerator of 1+1/prime(1)^2+ ... + 1/prime(n)^2 where prime(k) is the k-th prime.

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A115963 Numerator of Sum_{i=1..n} 1/prime(i)^3.

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A241189 Numerator of Sum_{i=1..n} 1/(prime(i)*prime(i+1)).

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A075987 Numerator(1+1/prime(1)^3+ ... + 1/prime(n)^3) where prime(k) is the k-th prime.

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