A112968 -id:A112968 - cp's OEIS frontend

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

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

Reinhard Zumkeller, Oct 07 2005

			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.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

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

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

Reinhard Zumkeller, Oct 07 2005

sign

			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.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

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

Reinhard Zumkeller, Oct 07 2005

sign

			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.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

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

a(n)=sum(i=1,n-1,moebius(i)*sigma(n-i)) \\ Charles R Greathouse IV, Feb 14 2013

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

Reinhard Zumkeller, Oct 07 2005

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A112965, A112968, A022559.

X:= Vector(100, numtheory:-bigomega):
seq(add(X[i]*X[n-i],i=1..n-1),n=1..100); # Robert Israel, Mar 15 2017

Table[Sum[PrimeOmega[i] PrimeOmega[n - i],{i,1, n - 1} ], {n, 1, 61}] (* Indranil Ghosh, Mar 16 2017 *)

for(n=1, 61, print1(sum(i=1, n - 1, bigomega(i) * bigomega(n - i)),", ")) \\ Indranil Ghosh, Mar 16 2017

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

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

Reinhard Zumkeller, Oct 07 2005

sign

			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.

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

with(numtheory); f:=n->add(phi(i)*mobius(n-i),i=1..n-1);

a(n)=sum(i=1,n-1,moebius(i)*eulerphi(n-i)) \\ Charles R Greathouse IV, Feb 21 2013

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

cp's OEIS Frontend

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

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Haskell

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

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Haskell

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

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PARI

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

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

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