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 1-6 of 6 results.

A330596 Decimal expansion of Product_{primes p} (1 - 1/p^2 + 1/p^3).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Dec 19 2019

Keywords

Comments

The asymptotic density of A337050. - Amiram Eldar, Aug 13 2020

Examples

			0.748535259682363564644215048637910601641640343005324404515852793925925...

Crossrefs

Cf. A057723, A065464, A065473, A065476, A065487, A328017, A330594, A330595, A337050, A372636.

Programs

  • Mathematica
    Do[Print[N[Exp[-Sum[q = Expand[(p^2 - p^3)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 110]], {t, 20, 200, 20}]
  • PARI
    prodeulerrat(1 - 1/p^2 + 1/p^3) \\ Amiram Eldar, Mar 17 2021

Formula

Equals (6/Pi^2) * A065487. - Amiram Eldar, Jun 10 2020

A330595 Decimal expansion of Product_{primes p} (1 + 1/p^2 + 1/p^3).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Dec 19 2019

Keywords

Examples

			1.748932997843245303033906997685114802259883493595480897273662144084849...

Crossrefs

Cf. A065464, A065473, A065476, A160889, A328017, A330594, A330596, A338325.

Programs

  • Mathematica
    Do[Print[N[Exp[-Sum[q = Expand[(-p^2 - p^3)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 110]], {t, 20, 200, 20}]
  • PARI
    prodeulerrat(1 + 1/p^2 + 1/p^3) \\ Vaclav Kotesovec, Sep 19 2020

Formula

Equals Sum_{n>=1} 1/A338325(n). - Amiram Eldar, Oct 26 2020

A107759 a(n) = (+2)UnitarySigma(n): if n = Product p_i^r_i then a(n) = Product (2 + p_i^r_i).

Original entry on oeis.org

1, 4, 5, 6, 7, 20, 9, 10, 11, 28, 13, 30, 15, 36, 35, 18, 19, 44, 21, 42, 45, 52, 25, 50, 27, 60, 29, 54, 31, 140, 33, 34, 65, 76, 63, 66, 39, 84, 75, 70, 43, 180, 45, 78, 77, 100, 49, 90, 51, 108, 95, 90, 55, 116, 91
Offset: 1

Views

Author

Yasutoshi Kohmoto, May 25 2005

Keywords

Examples

			a(12) = (2+3)*(2+4) = 30.

Links

Crossrefs

Cf. A007429, A034448, A054862, A072691, A107758, A330594.

Programs

  • Maple
    A107759 := proc(n) local pf,p ; if n = 1 then 1; else pf := ifactors(n)[2] ; mul( 2+op(1,p)^op(2,p), p=pf) ; end if; end proc:
seq(A107759(n),n=1..60) ; # R. J. Mathar, Jan 07 2011
  • Mathematica
    a[1] = 1; a[n_] := Times @@ (2 + Power @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 20 2020 *)

Formula

a(n) = Sum_{d|n, gcd(d, n/d) = 1} usigma(d), where usigma = A034448. - Ilya Gutkovskiy, Mar 27 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 1/p^2 - 2/p^3) = A072691 * A330594 = 0.910438... . - Amiram Eldar, Nov 01 2022

A208133 Total number of subgroups of rank <= 2 of a certain class of abelian groups of order n defined as direct products of Z/(mZ) X Z/(kZ) with m|k.

Original entry on oeis.org

1, 2, 2, 8, 2, 4, 2, 12, 9, 4, 2, 16, 2, 4, 4, 31, 2, 18, 2, 16, 4, 4, 2, 24, 11, 4, 14, 16, 2, 8, 2, 42, 4, 4, 4, 72, 2, 4, 4, 24, 2, 8, 2, 16, 18, 4, 2, 62, 13, 22, 4, 16, 2, 28, 4, 24, 4, 4, 2, 32, 2, 4, 18, 90, 4, 8, 2, 16, 4, 8, 2, 108, 2, 4, 22, 16
Offset: 1

Views

Author

R. J. Mathar, Mar 29 2012

Keywords

Comments

Level function l_tau^2(n) of Bhowmik and Wu.
Records occur at 1, 2, 4, 8, 12, 16, 32, 36, 64, 72, 108, 128, 144, 288, 432, 576, 1152, 1296, 2304, 3600, 5184, 7200, 9216, 10368, 14112, 14400, 20736, 28224, 28800, 32400, 57600, ... and they are: 1, 2, 8, 12, 16, 31, 42, 72, 90, 108, 112, 116, 279, 378, 434, 810, 1044, 1302, 2025, 3069, 3780, 4158, 4644, 4872, 4914, 8910, 9450, 10530, 11484, 14322, 22275, ... - Antti Karttunen, Mar 21 2018

References

  • A. Laurincikas, The universality of Dirichlet series attached to finite Abelian groups, in "Number Theory", Proc. Turku Sympos. on Number Theory, May 31-June 4, 1999, p 179.

Links

Crossrefs

Cf. A010052, A037213, A300828.

Programs

  • Maple
    L300828 := [ 1, 0, 0, 1, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0
] ;
L010052 := [ 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ];
L037213 := [ 1, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ;
Lx := DIRICHLET(L300828,L037213) ;
Lx := DIRICHLET(Lx,L010052) ;
Lx := DIRICHLET(Lx,L010052) ;
Lx := MOBIUSi(Lx) ;
Lx := MOBIUSi(Lx) ;
# Name of initial list L1 changed to L300828 to refer to sequence A300828 by Antti Karttunen, Mar 21 2018
  • PARI
    A037213(n) = if(issquare(n),sqrtint(n),0);
A300828(n) = { if(1==n, return(1)); my(val=1, v=factor(n), d=matsize(v)[1]); for(i=1,d, if(v[i,2] < 2 || v[i,2] > 3, return(0)); if (v[i,2] == 3, val *= -2)); return(val); };
a208133s1(n) = sumdiv(n,d,A037213(n/d)*A300828(d));
a208133s2(n) = sumdiv(n,d,issquare(n/d)*a208133s1(d));
a208133s3(n) = sumdiv(n,d,issquare(n/d)*a208133s2(d));
a208133s4(n) = sumdiv(n,d,a208133s3(d));
A208133(n) = sumdiv(n,d,a208133s4(d)); \\ Antti Karttunen, Mar 21 2018, after R. J. Mathar's Maple code
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 + X + 2*X^2)/(1 - X)^3/(1 + X)^2/(1 - p*X^2))[n], ", ")) \\ Vaclav Kotesovec, Jun 18 2020

Formula

Dirichlet g.f.: zeta(s)^2*zeta(2s)^2*zeta(2s-1)*Product_{primes p} (1 + 1/p^(2s) - 2/p^(3s)).
Sum_{k=1..n} a(k) ~ c * Pi^4 * log(n)^2 * n / 144, where c = A330594 = Product_{primes p} (1 + 1/p^2 - 2/p^3) = 1.10696011195321767665117913000743959294954883365812241904313404497877733241... - Vaclav Kotesovec, Dec 18 2019
More precise asymptotics: Let f(s) = Product_{primes p} (1 + 1/p^(2*s) - 2/p^(3*s)), then Sum_{k=1..n} a(k) ~ n*Pi^2 * (Pi^2 * f(1) * log(n)^2 + 2*Pi^2 * log(n) * (f(1) * (-1 + 8*gamma - 48*log(A) + 4*log(2*Pi)) + f'(1)) + Pi^2 * (2*f(1)*(1 + 25*gamma^2 + 576*log(A)^2 + log(A) * (48 - 96*log(2*Pi)) - 8*gamma * (1 + 36*log(A) - 3*log(2*Pi)) - 4*log(2*Pi) + 4*log(2*Pi)^2 - 6*sg1) + 2*(-1 + 8*gamma - 48*log(A) + 4*log(2*Pi))*f'(1) + f''(1)) + 48*f(1)*zeta''(2)) / 144, where f(1) = A330594, f'(1) = A330594 * (-A335705) = f(1) * Sum_{primes p} = -2*(p-3) * log(p) / (p^3 + p - 2) = -0.087825458097278818094375273108270679512035928574..., f''(1) = A330594 * (A335705^2 + A335706) = f'(1)^2/f(1) + f(1) * Sum_{primes p} = 2*p*(2*p^3 - 9*p^2 - 1) * log(p)^2) / (p^3 + p - 2)^2 = 0.26722508718782634450711076996710402451611235402675360769..., zeta''(2) = A201994, A is the Glaisher-Kinkelin constant A074962, gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Jun 18 2020

A382419 The product of exponents in the prime factorization of the cubefree numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 4, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Mar 25 2025

Keywords

Comments

Differs from A368712 at n = 1, 31, 85, 151, 164, 189, ... .
All the terms are powers of 2.

Links

Crossrefs

Cf. A002117, A004709, A005361, A330594, A368712, A376366, A382421, A382422.

Programs

  • Mathematica
    s[n_] := Times @@ FactorInteger[n][[;; , 2]]; cubeFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] < 3; s /@ Select[Range[120], cubeFreeQ]
  • PARI
    list(kmax) = {my(e); print1(1, ", "); for(k = 2, kmax, e = factor(k)[, 2]; if(vecmax(e) < 3, print1(vecprod(e), ", "))); }

Formula

a(n) = A005361(A004709(n)).
a(n) = 2^A376366(n).
a(n) >= A368712(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3) * Product_{p prime} (1 + 1/p^2 - 2/p^3) = A002117 * A330594 = 1.33062904409568262931... .

A382421 The product of exponents in the prime factorization of the noncubefree numbers.

Original entry on oeis.org

3, 4, 3, 3, 5, 3, 4, 3, 3, 6, 6, 4, 4, 3, 5, 3, 6, 4, 3, 3, 7, 3, 3, 8, 3, 5, 4, 3, 4, 3, 3, 6, 6, 4, 9, 5, 3, 4, 5, 3, 3, 8, 3, 3, 4, 3, 10, 3, 3, 4, 3, 6, 8, 3, 4, 3, 3, 3, 5, 6, 4, 3, 3, 3, 7, 6, 8, 4, 3, 5, 3, 12, 3, 6, 3, 3, 4, 3, 5, 5, 3, 4, 6, 6, 9, 3, 3
Offset: 1

Views

Author

Amiram Eldar, Mar 25 2025

Keywords

Links

Crossrefs

Cf. A002117, A013661, A013664, A005361, A046099, A082695, A330594, A368039, A382419.

Programs

  • Mathematica
    s[n_] := Times @@ FactorInteger[n][[;; , 2]]; noncubeFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] > 2; s /@ Select[Range[600], noncubeFreeQ]
  • PARI
    list(kmax) = {my(e); for(k = 2, kmax, e = factor(k)[, 2]; if(vecmax(e) > 2, print1(vecprod(e), ", "))); }

Formula

a(n) = A005361(A046099(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)*zeta(3)^2/zeta(6) - zeta(3) * Product_{p prime} (1 + 1/p^2 - 2/p^3))/(zeta(3) - 1) = (A082695 - A330594) * A002117 / (A002117 - 1) = 4.97723390794900554553... .
Showing 1-6 of 6 results.

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