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 1-3 of 3 results.

A360997 Multiplicative with a(p^e) = e + 3.

Original entry on oeis.org

1, 4, 4, 5, 4, 16, 4, 6, 5, 16, 4, 20, 4, 16, 16, 7, 4, 20, 4, 20, 16, 16, 4, 24, 5, 16, 6, 20, 4, 64, 4, 8, 16, 16, 16, 25, 4, 16, 16, 24, 4, 64, 4, 20, 20, 16, 4, 28, 5, 20, 16, 20, 4, 24, 16, 24, 16, 16, 4, 80, 4, 16, 20, 9, 16, 64, 4, 20, 16, 64, 4, 30, 4, 16
Offset: 1

Views

Author

Vaclav Kotesovec, Feb 28 2023

Keywords

Links

Crossrefs

Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), this sequence (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), A360909 (3*e+2), A360911 (3*e-2), A322328 (4*e), A360996 (5*e).
Cf. A000005, A007425, A007426, A074816.

Programs

  • Mathematica
    g[p_, e_] := e+3; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1+2*X-2*X^2)/(1-X)^2)[n], ", "))

Formula

Dirichlet g.f.: Product_{primes p} (1 + (4*p^s - 3)/(p^s - 1)^2).
Dirichlet g.f.: zeta(s)^4 * Product_{primes p} (1 - 5/p^(2*s) + 6/p^(3*s) - 2/p^(4*s)).
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = A000005(A361264(n)).
a(n) = A074816(n)*A007426(n)/A007425(n). (End)

A361266 Multiplicative with a(p^e) = p^(e + 3), e > 0.

Original entry on oeis.org

1, 16, 81, 32, 625, 1296, 2401, 64, 243, 10000, 14641, 2592, 28561, 38416, 50625, 128, 83521, 3888, 130321, 20000, 194481, 234256, 279841, 5184, 3125, 456976, 729, 76832, 707281, 810000, 923521, 256, 1185921, 1336336, 1500625, 7776, 1874161, 2085136, 2313441, 40000
Offset: 1

Views

Author

Vaclav Kotesovec, Mar 06 2023

Keywords

Comments

In general, if the function is multiplicative with a(p^e) = p^(e + m) where m > 0, then Dirichlet g.f.: Product_{primes p} (1 + p^(m+1)/(p^s - p)).
Equivalently, Dirichlet g.f.: zeta(s-m-1) * zeta(s-1) * Product_{primes p} (1 + p^(2 + m - 2*s) - p^(2 + 2*m - 2*s) - p^(1 - s)).
Sum_{k=1..n} a(k) ~ c(m) * zeta(m+1) * n^(m+2) / (m+2), where c(m) = Product_{primes p} (1 - 1/p^2 - 1/p^(m+1) + 1/p^(m+2)).
Limit_{m->oo} c(m) = 6/Pi^2 = A059956.

Links

Crossrefs

Cf. A003557, A007947, A064549, A361264.

Programs

  • Maple
    f:= proc(n) local t;
  mul(t[1]^(t[2]+3), t = ifactors(n)[2])
end proc:
map(f, [$1..50]); # Robert Israel, Mar 07 2023
  • Mathematica
    g[p_, e_] := p^(e+3); a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, 1 + p^4*X/(1 - p*X))[n], ", "))
  • PARI
    a(n) = my(f=factor(n)); for (k=1, #f~, f[k,2] +=3); factorback(f); \\ Michel Marcus, Mar 07 2023

Formula

Dirichlet g.f.: Product_{primes p} (1 + p^4/(p^s - p)).
Dirichlet g.f.: zeta(s-4) * zeta(s-1) * Product_{primes p} (1 + p^(5 - 2*s) - p^(8 - 2*s) - p^(1 - s)).
Sum_{k=1..n} a(k) ~ c * Pi^4 * n^5 / 450, where c = Product_{primes p} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.5761527353856670595206110782641172754062471168028961885...
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = n * A007947(n)^3 = A064549(n) * A007947(n)^2 = A361264(n) * A007947(n) = A064549(A064549(A064549(n))).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^3*(p-1))) = 1.148846213921... . (End)

A361268 Multiplicative with a(p^e) = e * p^(e + 2), e > 0.

Original entry on oeis.org

1, 8, 27, 32, 125, 216, 343, 96, 162, 1000, 1331, 864, 2197, 2744, 3375, 256, 4913, 1296, 6859, 4000, 9261, 10648, 12167, 2592, 1250, 17576, 729, 10976, 24389, 27000, 29791, 640, 35937, 39304, 42875, 5184, 50653, 54872, 59319, 12000, 68921, 74088, 79507, 42592
Offset: 1

Views

Author

Vaclav Kotesovec, Mar 06 2023

Keywords

Comments

In general, if the function is multiplicative with a(p^e) = e*p^(e+m) where m > 0, then Dirichlet g.f.: Product_{primes p} (1 + p^(s + m + 1)/(p^s - p)^2).
Equivalently, Dirichlet g.f.: zeta(s-m-1) * zeta(s-1)^2 * Product_{primes p} (1 - p^(3+m-3*s) + p^(2-2*s) + 2*p^(2+m-2*s) - p^(2+2*m-2*s) - 2*p^(1-s)).
Sum_{k=1..n} a(k) ~ c(m) * zeta(m+1)^2 * n^(m+2) / (m+2), where c(m) = Product_{primes p} (1 - 1/p^2 - 1/p^(2*m+3) + 1/p^(2*m+2) + 2/p^(m+2) - 2/p^(m+1)).
Limit_{m->oo} c(m) = 6/Pi^2 = A059956.

Links

Crossrefs

Cf. A005361, A059956, A203639, A361264, A361265.

Programs

  • Mathematica
    g[p_, e_] := e*p^(e+2); a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, 1 + p^3 * X / (1 - p*X)^2)[n], ", "))

Formula

Dirichlet g.f.: Product_{primes p} (1 + p^(s + 3)/(p^s - p)^2).
Dirichlet g.f.: zeta(s-3) * zeta(s-1)^2 * Product_{primes p} (1 - p^(5 - 3*s) + p^(2 - 2*s) + 2*p^(4 - 2*s) - p^(6 - 2*s) - 2*p^(1 - s)).
Sum_{k=1..n} a(k) ~ c * zeta(3)^2 * n^4 / 4, where c = Product_{primes p} (1 - 1/p^2 - 2/p^3 + 2/p^4 + 1/p^6 - 1/p^7) = 0.47448576370894461600229128319633117903859559137234612880645471185501089953...
a(n) = A005361(n) * A361264(n).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 - log(1-1/p))/p^2 = 1.24331517732028787738... . - Amiram Eldar, Sep 01 2023
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.