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.

A112966 Sum(mu(i)*omega(j): i+j=n), with mu=A008683 and omega=A001221.

Original entry on oeis.org

0, 0, 1, 0, -1, -1, -1, -2, -3, -2, -2, -1, -3, -3, -4, 0, -6, -1, -4, -1, -3, -3, -7, 0, -5, -3, -3, 0, -5, 1, -5, -2, -10, -1, -8, 4, -8, -3, -4, 2, -6, 0, -5, -1, -4, -2, -11, 3, -8, -1, -8, -1, -11, 2, -8, 2, -7, -3, -9, 5, -2, -5, -7, 2, -11, 7, -6, 0, -4, 1, -9, 4, -12, -3, -6, 0, -10, 2, -7, -1, -10, -8, -12, 6, -13, -2, -12, 0
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 07 2005

Keywords

Examples

			a(5)=mu(1)*omega(4)+mu(2)*omega(3)+mu(3)*omega(2)+mu(4)*omega(1)
= 1*1 - 1*1 - 1*1 + 0*0 = -1.

Links

Crossrefs

Cf. A013939, A112965, A068341, A112962, A112963, A112964, A112968.

Programs

  • Haskell
    a112966 n = sum $ zipWith (*)
   a008683_list $ reverse $ take (n - 1) a001221_list
-- Reinhard Zumkeller, Feb 29 2012

A112967 Sum(Omega(i)*Omega(j): i+j=n), with Omega=A001222.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 6, 10, 10, 17, 18, 26, 24, 33, 30, 41, 38, 52, 46, 64, 54, 71, 62, 87, 70, 91, 80, 106, 90, 116, 100, 130, 112, 139, 120, 163, 130, 161, 144, 185, 152, 190, 162, 208, 172, 205, 178, 244, 186, 232, 208, 262, 212, 267, 218, 291, 246, 287, 248, 329, 252
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 07 2005

Keywords

Links

Crossrefs

Cf. A112965, A112968, A022559.

Programs

  • Maple
    X:= Vector(100, numtheory:-bigomega):
seq(add(X[i]*X[n-i],i=1..n-1),n=1..100); # Robert Israel, Mar 15 2017
  • Mathematica
    Table[Sum[PrimeOmega[i] PrimeOmega[n - i],{i,1, n - 1} ], {n, 1, 61}] (* Indranil Ghosh, Mar 16 2017 *)
  • PARI
    for(n=1, 61, print1(sum(i=1, n - 1, bigomega(i) * bigomega(n - i)),", ")) \\ Indranil Ghosh, Mar 16 2017

Formula

G.f.: (Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)))^2. - Ilya Gutkovskiy, Mar 15 2017

A300671 Expansion of 1/(1 - Sum_{k>=1} x^prime(k)/(1 - x^prime(k))).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 8, 15, 23, 40, 63, 108, 172, 290, 471, 782, 1280, 2119, 3474, 5741, 9432, 15557, 25590, 42180, 69413, 114371, 188276, 310136, 510637, 841045, 1384883, 2280831, 3755862, 6185457, 10185941, 16774695, 27624215, 45492412, 74916559, 123374127, 203172520, 334587577
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 11 2018

Keywords

Comments

Invert transform of A001221.

Links

Crossrefs

Cf. A001221, A013939, A112965, A129921, A180305, A293548, A300672.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*nops(ifactors(i)[2]), i=1..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    nmax = 42; CoefficientList[Series[1/(1 - Sum[x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 42; CoefficientList[Series[1/(1 - Sum[PrimeNu[k] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PrimeNu[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 42}]

Formula

G.f.: 1/(1 - Sum_{k>=2} A001221(k)*x^k).
Showing 1-3 of 3 results.

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