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 1-5 of 5 results.

A112968 a(n) = Sum_{i+j=n} mu(i)*Omega(j), with mu=A008683 and Omega=A001222.

Original entry on oeis.org

0, 0, 1, 0, 0, -2, -2, -2, -2, -2, -6, -2, -4, -2, -7, -1, -5, 0, -7, -3, -9, 1, -11, 2, -7, 1, -12, 1, -11, 7, -8, -5, -8, -1, -18, 3, -10, 1, -13, 1, -7, 13, -12, -2, -13, 6, -16, 3, -11, 3, -15, -4, -16, 13, -15, -4, -15, 4, -17, 11, -14, 4, -13, 7, -12, 15, -17, -5, -15, 16, -13, 3, -12, 3, -20, 3, -27, 19, -20, -3, -11, 3
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*2 - 1*1 - 1*1 + 0*1 = 0.

Links

Crossrefs

Cf. A013939, A112967, A068341, A112962, A112963, A112964, A112966.

Programs

  • Haskell
    a112968 n = sum $ zipWith (*)
   a008683_list $ reverse $ take (n - 1) a001222_list
-- Reinhard Zumkeller, Feb 29 2012
  • Mathematica
    A112968[n_]:=Plus@@Table[MoebiusMu[i]*PrimeOmega[n-i],{i,1,n-1}]; Array[A112968,200] (* Enrique Pérez Herrero, Feb 28 2012 *)

Extensions

Corrected by N. J. A. Sloane, Mar 01 2006

A112963 Sum(mu(i)*tau(j): i+j=n), with mu=A008683 and tau=A000005.

Original entry on oeis.org

0, 1, 1, -1, -1, -4, -2, -5, -5, -4, -7, -4, -6, -7, -11, 0, -12, -1, -11, -6, -12, -1, -20, 2, -13, -2, -16, 2, -19, 9, -18, -9, -20, 4, -31, 10, -21, -2, -18, 7, -20, 14, -26, -3, -16, 13, -40, 5, -26, 7, -22, -1, -40, 18, -32, 2, -21, 10, -40, 16, -25, 5, -21, 17, -41, 31, -40, -4, -14, 30, -38, 3, -39, 8, -21, 14, -58
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 07 2005

Keywords

Examples

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

Links

Crossrefs

Cf. A006218, A055507, A068341, A112962, A112964, A112966, A112968.

Programs

  • Haskell
    a112963 n = sum $ zipWith (*)
   a008683_list $ reverse $ take (n - 1) a000005_list
-- Reinhard Zumkeller, Feb 29 2012

A112964 Sum(mu(i)*sigma(j): i+j=n), with mu=A008683 and sigma=A000203.

Original entry on oeis.org

0, 1, 2, 0, 0, -6, -3, -12, -11, -13, -22, -19, -20, -30, -41, -15, -55, -24, -52, -41, -59, -24, -109, -22, -78, -42, -111, -26, -131, -2, -119, -75, -133, -8, -214, 7, -175, -68, -176, -17, -209, 14, -231, -73, -175, 45, -349, -11, -236, -20, -236, -53, -384, 68, -321, -56, -270, 1, -457, 41, -328, -48
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 07 2005

Keywords

Examples

			a(5)=mu(1)*sigma(4)+mu(2)*sigma(3)+mu(3)*sigma(2)+mu(4)*sigma(1)
= 1*7 - 1*4 - 1*3 + 0*1 = 0.

Links

Crossrefs

Cf. A024916, A000385, A068341, A112962, A112963, A112966, A112968.

Programs

A112962 Sum(mu(i)*phi(j): i+j=n), with mu=A008683 and phi=A000010.

Original entry on oeis.org

0, 1, 0, 0, -1, -1, -4, -2, -5, -8, -5, -8, -9, -11, -10, -24, 1, -21, -11, -23, -15, -37, 4, -42, -11, -38, -7, -49, 6, -63, -12, -44, -3, -81, 10, -106, 7, -49, -8, -92, 15, -103, 2, -72, -5, -114, 41, -140, -3, -114, 8, -113, 49, -179, 3, -135, 27, -131, 46, -218, -7, -99, 32, -185, 72, -259, 50, -104, 23, -211, 52, -248, 43, -153
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 07 2005

Keywords

Examples

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

Crossrefs

Cf. A002088, A065093, A068341, A112963, A112964, A112966, A112968.

Programs

  • Maple
    with(numtheory); f:=n->add(phi(i)*mobius(n-i),i=1..n-1);
  • PARI
    a(n)=sum(i=1,n-1,moebius(i)*eulerphi(n-i)) \\ Charles R Greathouse IV, Feb 21 2013

Extensions

Corrected by N. J. A. Sloane, Mar 01 2006

A112965 a(n) = Sum_{i+j=n} omega(i)*omega(j), where omega = A001221.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 4, 7, 8, 9, 10, 14, 14, 17, 18, 23, 24, 27, 26, 32, 32, 35, 36, 44, 42, 47, 48, 52, 50, 58, 54, 65, 62, 67, 66, 78, 70, 79, 78, 88, 84, 94, 88, 100, 100, 103, 100, 118, 106, 119, 114, 124, 116, 135, 122, 138, 134, 141, 136, 155, 142, 155, 154, 163, 156
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 07 2005

Keywords

Links

Crossrefs

Cf. A013939, A112966, A112967.

Programs

  • Mathematica
    Table[Sum[PrimeNu[i]*PrimeNu[n - i], {i, n - 1}], {n, 65}] (* Ivan Neretin, Jan 21 2017 *)
  • PARI
    a(n) = sum(i=2, n-2, omega(i)*omega(n-i)); \\ Michel Marcus, Jan 22 2017

Formula

G.f.: (Sum_{k>=1} x^prime(k)/(1 - x^prime(k)))^2. - Ilya Gutkovskiy, Jan 31 2017
Showing 1-5 of 5 results.

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

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