A106315 Harmonic residue of n.

0, 1, 2, 5, 4, 0, 6, 2, 1, 4, 10, 16, 12, 8, 12, 18, 16, 30, 18, 36, 20, 16, 22, 12, 13, 20, 28, 0, 28, 24, 30, 3, 36, 28, 44, 51, 36, 32, 44, 50, 40, 48, 42, 12, 36, 40, 46, 108, 33, 21, 60, 18, 52, 72, 4, 88, 68, 52, 58, 48, 60, 56, 66, 67, 8, 96, 66, 30, 84, 128, 70, 84, 72, 68, 78

Offset: 1

Author

George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Apr 29 2005

Keywords

• nonn

Comments

The harmonic residue is the remainder when n*d(n) is divided by sigma(n), where d(n) is the number of divisors of n and sigma(n) is the sum of the divisors of n. If n is perfect, the harmonic residue of n is 0.

Links

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

Crossrefs

Cf. A106316, A106317, A001599 (positions of zeros).

Cf. A000005, A000203.

Programs

Haskell

a106315 n = n * a000005 n `mod` a000203 n -- Reinhard Zumkeller, Apr 06 2014

Maple

A106315 := proc(n)
    modp(n*numtheory[tau](n),numtheory[sigma](n)) ;
end proc:
seq(A106315(n),n=1..100) ; # R. J. Mathar, Jan 25 2017

Mathematica

HarmonicResidue[n_]=Mod[n*DivisorSigma[0, n], DivisorSigma[1, n]]; HarmonicResidue[ Range[ 80]]

Formula

a(n) = A038040(n) - A000203(n) * A240471(n) . - Reinhard Zumkeller, Apr 06 2014

Extensions

Mathematica program completed by Harvey P. Dale, Feb 29 2024