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 1-8 of 8 results.

A086242 Decimal expansion of the sum of 1/(p-1)^2 over all primes p.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Jul 13 2003

Keywords

Examples

			1.37506499474863528791725313052243969917959996017...

References

  • Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, pp. 94-98.

Links

Crossrefs

Cf. A000010, A007434, A119686, A334746, A382554.

Programs

  • Mathematica
    digits = 116; Np = NSum[(n-1)*PrimeZetaP[n], {n, 2, Infinity}, NSumTerms -> 3*digits, WorkingPrecision -> digits+10]; RealDigits[Np, 10, digits] // First (* Jean-François Alcover, Sep 02 2015 *)
  • PARI
    default(realprecision,256);
(f(k)=return(sum(n=1,1024,moebius(n)/n*log(zeta(k*n)))));
sum(k=2,1024,(k-1)*f(k)) /* Robert Gerbicz, Sep 12 2012 */
  • PARI
    sumeulerrat(1/(p-1)^2) \\ Amiram Eldar, Mar 19 2021

Formula

Equals Sum_{k>=2} (k-1)*primezeta(k). - Robert Gerbicz, Sep 12 2012
Equals lim_{n -> oo} A119686(n)/A334746(n). - Petros Hadjicostas, May 11 2020
Equals Sum_{k>=2} (J_2(k)-phi(k)) * log(zeta(k)) / k, where J_2 = A007434 and phi = A000010 (Jakimczuk, 2017). - Amiram Eldar, Mar 18 2024

Extensions

More digits copied from Cohen's paper by R. J. Mathar, Dec 05 2008
More terms from Robert Gerbicz, Sep 12 2012

A120271 a(n) = numerator(Sum_{k=1..n} 1/(prime(k)-1)).

Original entry on oeis.org

1, 3, 7, 23, 121, 21, 173, 1597, 17927, 127469, 129317, 43619, 44081, 44521, 1033223, 13538159, 395369371, 132680013, 400467919, 402757063, 1214947859, 1221110939, 50305908619, 50529880549, 101470376303, 509322834499, 8691337402883
Offset: 1

Views

Author

Alexander Adamchuk, Jul 01 2006

Keywords

Comments

a(n) is squarefree except for n = 5, 14, 49, ... where squared prime factors are 11, 211, 479, ...
a(n)/A128646(n) is the asymptotic mean over the positive integers of the number of prime divisors that are not greater than prime(n), counted with multiplicity (cf. A007814, A169611, A356006). - Amiram Eldar, Jul 23 2022

Links

Crossrefs

Cf. A128646 (denominators), A119686, A006093, A000040.
Cf. A135212, A078456.
Cf. A007814, A169611, A356006.

Programs

  • Maple
    R:= [seq(1/(ithprime(k)-1),k=1..40)]:
S:= ListTools:-PartialSums(R):
A:= map(numer,S); # Robert Israel, Jan 12 2025
  • Mathematica
    Numerator[Table[Sum[1/(Prime[i]-1),{i,1,n}],{n,1,50}]]
Accumulate[1/(Prime[Range[30]]-1)]//Numerator (* Harvey P. Dale, May 03 2025 *)
  • PARI
    a(n) = numerator(sum(k=1, n, 1/(prime(k)-1))); \\ Michel Marcus, Oct 02 2016

Formula

a(n) = numerator(Sum_{k=1..n} 1/(prime(k)-1)).
a(n) = A078456(n) * A135212(n). - Alexander Adamchuk, Nov 23 2007

A128646 a(n) = denominator(Sum_{k=1..n} 1/(prime(k)-1)).

Original entry on oeis.org

1, 2, 4, 12, 60, 10, 80, 720, 7920, 55440, 55440, 18480, 18480, 18480, 425040, 5525520, 160240080, 53413360, 160240080, 160240080, 480720240, 480720240, 19709529840, 19709529840, 39419059680, 197095298400, 3350620072800
Offset: 1

Views

Author

Alexander Adamchuk, Mar 18 2007

Keywords

Comments

A120271(n) = numerator(Sum_{k=1..n} 1/(prime(k)-1)); A128648(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1)); numbers m such that a(m) = A128648(m) are listed in A128649.

Links

  • Eric Weisstein's World of Mathematics, Prime Sums

Crossrefs

Cf. A120271 (numerator(Sum_{k=1..n} 1/(prime(k)-1))).
Cf. A128649, A128647, A128648 (denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1))).
Cf. A119686, A006093, A000040.

Programs

  • Mathematica
    Table[Denominator[Sum[1/(Prime[k]-1),{k,1,n}]],{n,1,36}]

Formula

a(n) = denominator(Sum_{k=1..n} 1/(prime(k)-1)).

A128647 a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1)).

Original entry on oeis.org

1, 1, 3, 7, 41, 3, 53, 437, 5167, 34189, 36037, 3833, 3987, 11521, 274223, 3458639, 103063291, 100392623, 34273501, 33510453, 308270747, 302107667, 12626774467, 12402802537, 25216220279, 124110148411, 2142721739387, 111888942151111
Offset: 1

Views

Author

Alexander Adamchuk, Mar 18 2007

Keywords

Comments

Numbers m such that A128648(m) = A128646(n) are listed in A128649.

Links

  • Eric Weisstein's World of Mathematics, Prime sums

Crossrefs

Cf. A128648 (denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1))).
Cf. A120271 (numerator(Sum_{k=1..n} 1/(prime(k)-1))).
Cf. A128646, A128649, A119686, A006093, A000040.

Programs

  • Mathematica
    Table[Numerator[Sum[(-1)^(k+1)*1/(Prime[k]-1),{k,1,n}]],{n,1,36}]

Formula

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1)).

A128648 a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1)).

Original entry on oeis.org

1, 2, 4, 12, 60, 5, 80, 720, 7920, 55440, 55440, 6160, 6160, 18480, 425040, 5525520, 160240080, 160240080, 53413360, 53413360, 480720240, 480720240, 19709529840, 19709529840, 39419059680, 197095298400, 3350620072800, 177582863858400
Offset: 1

Views

Author

Alexander Adamchuk, Mar 18 2007

Keywords

Comments

Numbers m such that a(m) equals A128646(m) are listed in A128649.

Links

  • Eric Weisstein's World of Mathematics, Prime Sums

Crossrefs

Cf. A128647 (numerator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1))).
Cf. A128646 (denominator(Sum_{k=1..n} 1/(prime(k)-1))).
Cf. A128649, A120271, A119686, A006093, A000040.

Programs

  • Mathematica
    Table[Denominator[Sum[(-1)^(k+1)*1/(Prime[k]-1),{k,1,n}]],{n,1,36}]

Formula

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1)).

A128649 Numbers m such that A128646(m) = A128648(m).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 14, 15, 16, 17, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 65, 66, 71, 72, 73, 74, 75, 76, 77, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 539, 540, 541, 542, 543, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610
Offset: 1

Views

Author

Alexander Adamchuk, Mar 18 2007

Keywords

Comments

Terms of this sequence are 1..5, 7..11, 14..17, 21..35, 65..66, 71..77, 81..93, 539..543, 600..639, 644..650, 707..818, 1152..1185, 4502..4577, 4601..4823, 4893..5003, 7483..7633, ...

Links

  • Eric Weisstein's World of Mathematics, Prime Sums

Crossrefs

Cf. A128648 (denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1))).
Cf. A128646 (denominator(Sum_{k=1..n} 1/(prime(k)-1))).
Cf. A128647, A120271, A119686, A006093, A000040.

Programs

  • Mathematica
    f=0;g=0;Do[p=Prime[n];f=f+1/(p-1);g=g+(-1)^(n+1)*1/(p-1);kf=Denominator[f];kg=Denominator[g];If[Equal[kf,kg],Print[n]],{n,1,10000}]

A334746 Denominator of Sum_{k=1..n} 1/(prime(k) - 1)^2.

Original entry on oeis.org

1, 4, 16, 144, 3600, 1800, 57600, 518400, 62726400, 3073593600, 614718720, 614718720, 3073593600, 3073593600, 1625931014400, 274782341433600, 231091949145657600, 231091949145657600, 231091949145657600, 231091949145657600, 77030649715219200
Offset: 1

Views

Author

Petros Hadjicostas, May 11 2020

Keywords

Comments

Lim_{n -> infinity} A119686(n)/a(n) = A086242.

Examples

			The first few fractions are 1, 5/4, 21/16, 193/144, 4861/3600, 2443/1800, 78401/57600, 707209/518400, ... = A119686/A334746.

Links

  • Eric Weisstein's World of Mathematics, Prime Sums.

Crossrefs

Cf. A000040, A006093, A086242, A119686 (numerators).

Programs

  • Mathematica
    Denominator @ Accumulate @ Table[1/(Prime[k] - 1)^2, {k, 1, 21}] (* Amiram Eldar, May 12 2020 *)
  • PARI
    a(n) = denominator(sum(k=1, n, 1/(prime(k) - 1)^2)); \\ Michel Marcus, May 12 2020

A119723 Numerator of Sum[ (-1)^(k-1) * 1/(Prime[k]-1)^2, {k,1,n}].

Original entry on oeis.org

1, 3, 13, 113, 2861, 709, 45601, 408809, 49595489, 2426258561, 485934733, 485460413, 2429223061, 2427480661, 1284905668069, 217047437215261, 182605590283392901, 36508279615059377, 36518889897389297
Offset: 1

Views

Author

Alexander Adamchuk, Jun 13 2006

Keywords

Links

  • Eric Weisstein's World of Mathematics, Prime Sums.

Crossrefs

Cf. A119686, A006093, A000040, A086242.

Programs

  • Mathematica
    Numerator[Table[Sum[(-1)^(i-1)*1/(Prime[i]-1)^2, {i, 1, n}], {n, 1, 30}]]

Formula

a(n) = numerator[ Sum[ (-1)^(k-1) * 1/(Prime[k]-1)^2, {k,1,n}]].
Showing 1-8 of 8 results.

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