A210679 Number of distinct prime factors <= 7 of n.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 0, 2, 0, 2, 2, 1, 0, 2, 0, 2, 2, 1, 0, 2, 1, 1, 1, 2, 0, 3, 0, 1, 1, 1, 2, 2, 0, 1, 1, 2, 0, 3, 0, 1, 2, 1, 0, 2, 1, 2, 1, 1, 0, 2, 1, 2, 1, 1, 0, 3, 0, 1, 2, 1, 1, 2, 0, 1, 1, 3, 0, 2, 0, 1, 2, 1, 1, 2, 0, 2, 1, 1, 0, 3, 1, 1

Offset: 1

Author

Reinhard Zumkeller, Apr 01 2012

Keywords

• nonn
• easy

Comments

Periodic with period length 210. - Amiram Eldar, Sep 16 2023

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (-3,-5,-6,-6,-5,-3,0,3,5,6,6,5,3,1).

Crossrefs

Cf. A001221, A002473, A008364, A027748, A080672, A165743.

Number of distinct prime factors <= p: A171182 (p=3), A178146 (p=5), this sequence (p=7).

Programs

Haskell

a210679 = length . takeWhile (<= 7) . a027748_row

Mathematica

Join[{0},Table[Count[FactorInteger[n][[All,1]],?(#<8&)],{n,2,100}]] (* _Harvey P. Dale, Aug 18 2021 *)
a[n_] := PrimeNu[GCD[n, 210]]; Array[a, 100] (* Amiram Eldar, Sep 16 2023 *)

PARI

a(n) = omega(gcd(n, 210)); \\ Amiram Eldar, Sep 16 2023

Formula

a(n) <= 4.

a(A008364(n)) = 0; a(A080672(n)) > 0.

a(n) = A001221(n) iff n is 7-smooth: a(A002473(n)) = A001221(A002473(n)). [corrected by Amiram Eldar, Sep 16 2023]

From Amiram Eldar, Sep 16 2023: (Start)

Additive with a(p^e) = 1 if p <= 7, and 0 otherwise.

a(n) = A001221(A165743(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 247/210. (End)