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

1,3

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

Formula

a(n) = A038040(n) - A000203(n) * A240471(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]]

Prog

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

Crossrefs

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

Cf. A000005, A000203.

Sequence in context: A191474 A199602 A324057 * A285295 A217563 A373427

Adjacent sequences: A106312 A106313 A106314 * A106316 A106317 A106318

Keyword

nonn

Author

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

Extensions

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

Status

approved