A139396 -id:A139396 - cp's OEIS frontend

A007467 Product of next n primes.

2, 15, 1001, 215441, 95041567, 66238993967, 63009974049301, 87796770491685553, 173955570033393401009, 421385360593324054690769, 1172248885422611971256631487, 5253333091597988325086927419397, 21476254926032216698855019795863013

Offset: 1

N. J. A. Sloane, Simon Plouffe

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harvey P. Dale, Table of n, a(n) for n = 1..185
Ronald K. Hoeflin, Titan Test.

Cf. A139395, A139396, A238234.

terms=20;With[{prs=Prime[Range[(terms(terms+1))/2]]},Table[ Times@@ Take[prs,{(n(n-1))/2+1,(n(n+1))/2}],{n,terms}]] (* Harvey P. Dale, Aug 06 2013 *)
With[{nn=40},Times@@@TakeList[Prime[Range[(nn(nn+1))/2]],Range[nn]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Jan 15 2020 *)

a(n)=my(s=1);forprime(p=prime(n*(n-1)/2+1),prime(n*(n+1)/2),s*=p);s \\ Charles R Greathouse IV, Aug 06 2013

from math import prod
from sympy import prime
def a(n): return prod(prime(i) for i in range((n-1)*n//2+1, n*(n+1)//2+1))
print([a(n) for n in range(1, 14)]) # Michael S. Branicky, Feb 15 2021

From Amiram Eldar, Nov 15 2020: (Start)
Sum_{n>=1} 1/a(n) = A139395.
Sum_{n>=1} (-1)^(n+1)/a(n) = A238234 = 1 - A139396. (End)

Corrected and extended by Harvey P. Dale, Aug 06 2013

A139395 Decimal expansion of the sum 1/p(1) + 1/(p(2)p(3)) + 1/(p(4)p(5)*p(6)) + ..., where p(n) is the n-th prime.

Original entry on oeis.org

5, 6, 7, 6, 7, 0, 3, 1, 9, 8, 4, 4, 5, 1, 9, 7, 2, 0, 8, 4, 0, 6, 9, 8, 2, 4, 2, 2, 9, 4, 2, 3, 2, 6, 7, 2, 5, 5, 8, 1, 1, 4, 8, 1, 3, 7, 3, 4, 6, 7, 3, 7, 8, 6, 3, 8, 3, 4, 2, 4, 9, 4, 7, 0, 5, 4, 8, 5, 8, 5, 1, 8, 7, 8, 5, 8, 7, 8, 4, 5, 9, 7, 3, 8, 2, 4, 3, 4, 1, 1, 9, 8, 1, 2, 1, 4, 3, 8, 3, 7

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Apr 18 2008

Absolute difference between this number and A139396 is about 0.002.

			0.56767031984451972084069824229423267255811481373467378638342..

Cf. A139396.

P:=proc(n) local a,b,i,j,k; a:=0.5; k:=1; for i from 2 by 1 to n do b:=1; for j from k by 1 to k+i-1 do b:=b*1/ithprime(j+1); od; k:=j; a:=evalf(a+b,105); od; print(a); end: P(100);

With[{nn=100},RealDigits[Total[1/Times@@@TakeList[Prime[Range[(nn(nn+1))/2]],Range[nn]]],10,120][[1]]] (* Harvey P. Dale, Dec 20 2024 *)

A238234 Decimal expansion of the alternating sum 1/p(1) - 1/(p(2)p(3)) + 1/(p(4)p(5)p(6)) - 1/(p(7)p(8)p(9)p(10)) + ..., where p(n) is the n-th prime.

Original entry on oeis.org

4, 3, 4, 3, 2, 7, 7, 0, 3, 1, 9, 6, 9, 3, 8, 1, 0, 2, 2, 9, 6, 1, 5, 7, 5, 1, 3, 0, 2, 4, 8, 3, 7, 2, 3, 6, 7, 4, 2, 7, 9, 1, 3, 8, 9, 2, 7, 7, 1, 9, 6, 7, 7, 9, 3, 8, 5, 5, 2, 6, 0, 1, 4, 1, 4, 4, 2, 1, 1, 5, 0, 5, 4, 1, 6, 0, 9, 4, 6, 8, 0, 4, 0, 7, 3, 8, 9, 6, 1, 9, 8, 6, 8, 6, 1, 4, 2, 9, 1, 5, 2, 7, 8, 5, 7

Offset: 0

Paolo P. Lava, Feb 27 2014 - following a suggestion of Jean-François Alcover

Absolute difference between this number and A139395 is about 0.1333426...

			0.4343277031969381022961575130248372367427913892771967793855...

Cf. A139395, A139396.

P:=proc(n) local a, b, i, j, k; a:=0.5; k:=1; for i from 2 by 1 to n do b:=1; for j from k by 1 to k+i-1 do b:=b*1/ithprime(j+1); od; k:=j; a:=evalf(a+b*(-1)^(i-1), 105); od; print(a); end: P(100);

digits = 105; n0 = 10; dn = 10; t[n_] := n*(n + 1)/2; Clear[p]; p[n_] := p[n] = Sum[(-1)^(k + 1)/Product[Prime[j], {j, t[k] - k + 1, t[k]}], {k, 1, n}] // N[#, digits] &; p[n0]; p[n = n0 + dn]; While[RealDigits[p[n]] != RealDigits[p[n - dn]], Print["n = ", n]; n = n + dn]; RealDigits[p[n], 10, digits] // First (* Jean-François Alcover, Aug 12 2014, adapted from PARI *)

default(realprecision, 120);
T(n) = n*(n + 1)/2; \\ T(n) = A000217(n).
sum(k = 1, 100, (-1.)^(k-1)/prod(j = T(k) - k + 1, T(k), prime(j))) \\ Rick L. Shepherd, Mar 07 2014

More terms from and offset corrected by Rick L. Shepherd, Mar 07 2014

cp's OEIS Frontend

A007467 Product of next n primes.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A139395 Decimal expansion of the sum 1/p(1) + 1/(p(2)p(3)) + 1/(p(4)p(5)*p(6)) + ..., where p(n) is the n-th prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A238234 Decimal expansion of the alternating sum 1/p(1) - 1/(p(2)p(3)) + 1/(p(4)p(5)p(6)) - 1/(p(7)p(8)p(9)p(10)) + ..., where p(n) is the n-th prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

cp's OEIS Frontend

A007467 Product of next n primes.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A139395 Decimal expansion of the sum 1/p(1) + 1/(p(2)*p(3)) + 1/(p(4)*p(5)*p(6)) + ..., where p(n) is the n-th prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A238234 Decimal expansion of the alternating sum 1/p(1) - 1/(p(2)*p(3)) + 1/(p(4)*p(5)*p(6)) - 1/(p(7)*p(8)*p(9)*p(10)) + ..., where p(n) is the n-th prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A139395 Decimal expansion of the sum 1/p(1) + 1/(p(2)p(3)) + 1/(p(4)p(5)*p(6)) + ..., where p(n) is the n-th prime.

A238234 Decimal expansion of the alternating sum 1/p(1) - 1/(p(2)p(3)) + 1/(p(4)p(5)p(6)) - 1/(p(7)p(8)p(9)p(10)) + ..., where p(n) is the n-th prime.