A361384 a(n) is the number of distinct prime factors of the n-th unitary harmonic number.

0, 2, 2, 3, 3, 4, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 4, 5, 4, 4, 3, 5, 4, 5, 5, 5, 5, 5, 5, 4, 5, 5, 4, 5, 5, 5, 5, 4, 4, 4, 5, 6, 5, 6, 5, 5, 6, 6, 5, 5, 6, 6, 5, 5, 6, 6, 6, 6, 5, 6, 5, 6, 6, 6, 5, 6, 5, 6, 5, 6, 5, 6, 4, 5, 6, 6, 6, 6, 5, 6, 5, 6, 6, 6, 6, 5, 6

Offset: 1

Author

Amiram Eldar, Mar 10 2023

Keywords

• nonn

Comments

Each term appears a finite number of times in the sequence (Hagis and Lord, 1975).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..290
• Peter Hagis, Jr. and Graham Lord, Unitary harmonic numbers, Proc. Amer. Math. Soc., Vol. 51, No. 1 (1975), pp. 1-7.

Crossrefs

Cf. A001221, A006086, A006087.

Programs

Mathematica

uh[n_] := n * Times @@ (2/(1 + Power @@@ FactorInteger[n])); uh[1] = 1; PrimeNu[Select[Range[10^6], IntegerQ[uh[#]] &]]

PARI

uhmean(n) = {my(f = factor(n)); n*prod(i=1, #f~, 2/(1+f[i, 1]^f[i, 2])); };
lista(kmax) = {my(uh); for(k = 1, kmax, uh = uhmean(k); if(denominator(uh) == 1, print1(omega(k), ", ")));}

Formula

a(n) = A001221(A006086(n)).