A075986 -id:A075986 - cp's OEIS frontend

A136365 Numbers k such that A075986(k) is prime.

1, 171, 210, 550, 1445, 1809, 2176, 2719

Offset: 1

Alexander Adamchuk, Dec 27 2007

Cf. A075986 (numerator of 1 + 1/p(1)^2 + ... + 1/p(n)^2, where p(k) = prime(k)).
Cf. A024528, A125707.
Cf. A136366, A136367, A136368, A136369, A136370, A136371.

f=1; Do[ p=Prime[n]; f=f + 1/p^2; g=Numerator[ f ]; If[ PrimeQ[ g ], Print[ {n, g} ] ], {n, 1, 210} ]

a(4)-a(8) from Robert Price, Aug 26 2019

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

A061015 Numerator of Sum_{i=1..n} 1/p(i)^2, p(i) = i-th prime.

Original entry on oeis.org

1, 13, 361, 18589, 2293369, 392915461, 114454369129, 41578647715669, 22089188627685001, 18626778064527922741, 17942190650501641587001, 24603083510737933160021269, 41412850736015889039729489289, 76664929233749755566050236079461

Offset: 1

Winston C. Yang (winston(AT)cs.wisc.edu), May 21 2001

Michael S. Branicky, Table of n, a(n) for n = 1..195

Cf. A000040, A075986, A075987.

summ := 0: for n from 1 to 100 do if (isprime(n)) then summ := summ + 1/n^2; printf("%d,", numer(summ)); #printf("%d,", denom(summ)); end if; od; evalf(summ);

Numerator[Accumulate[1/Prime[Range[13]]^2]] (* Jayanta Basu, Jul 14 2013 *)

from sympy import prime
from fractions import Fraction
from itertools import accumulate, count, islice
def A061015gen(): yield from map(lambda x: x.numerator, accumulate(Fraction(1, prime(k)**2) for k in count(1)))
print(list(islice(A061015gen(), 20))) # Michael S. Branicky, Jun 26 2022

a(1) = 1; a(n) = a(n-1)*p(n)^2+(p(1)*...*p(n-1))^2. - Zak Seidov, Sep 28 2002

a(14) and beyond from Michael S. Branicky, Jun 26 2022

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.

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.

A131483 Meissel_Lehmer recursion: a(n,m) = a(n,m-1)-a(Floor[n/Prime[m]],m-1).

Original entry on oeis.org

1, 0, -1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1

Offset: 1

Roger L. Bagula, Oct 01 2007

			{1},
{0, -1},
{0, -1, -1},
{0, 0, 0, 0},
{0, 0, 0, 0, 0},
{0, 0, 1, 1, 1, 1},
{0, 0, 1, 1, 1, 1, 1},
{0, 0, 1, 1, 1, 1, 1, 1},
{0, 0, 1, 1, 1, 1, 1, 1, 1},
{0, 0, 1, 1, 1, 1, 1, 1, 1, 1},
{0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

J. C. Lagarias, V. S. Miller and A. M. Odlyzko, Computing pi(x): The Meissel-Lehmer method, Math. Comp., 44 (1985), pp. 537-560.

Cf. A000720, A006880, A007053, A075986, A059305.

a(1,1) = 1; a(n,m) = a(n,m-1)-a(Floor[n/Prime[m]],m-1);

cp's OEIS Frontend

A136365 Numbers k such that A075986(k) is prime.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A061015 Numerator of Sum_{i=1..n} 1/p(i)^2, p(i) = i-th prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A131483 Meissel_Lehmer recursion: a(n,m) = a(n,m-1)-a(Floor[n/Prime[m]],m-1).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula