A078898 Number of times the smallest prime factor of n is the smallest prime factor for numbers <= n; a(0)=0, a(1)=1.

0, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 9, 1, 10, 4, 11, 1, 12, 2, 13, 5, 14, 1, 15, 1, 16, 6, 17, 3, 18, 1, 19, 7, 20, 1, 21, 1, 22, 8, 23, 1, 24, 2, 25, 9, 26, 1, 27, 4, 28, 10, 29, 1, 30, 1, 31, 11, 32, 5, 33, 1, 34, 12, 35, 1, 36, 1, 37, 13, 38, 3, 39, 1, 40, 14, 41, 1, 42, 6, 43

Offset

0,5

Comments

From Antti Karttunen, Dec 06 2014: (Start)

For n >= 2, a(n) tells in which column of the sieve of Eratosthenes (see A083140, A083221) n occurs in. A055396 gives the corresponding row index.

(End)

Links

Harvey P. Dale (terms 1 - 1000) & Antti Karttunen, Table of n, a(n) for n = 0..10000

Formula

Ordinal transform of A020639 (Lpf). - Franklin T. Adams-Watters, Aug 28 2006

From Antti Karttunen, Dec 05-08 2014: (Start)

a(0) = 0, a(1) = 1, a(n) = 1 + a(A249744(n)).

a(0) = 0, a(1) = 1, a(n) = sum_{d | A002110(A055396(n)-1)} moebius(d) * floor(n / (A020639(n)*d)).

a(0) = 0, a(1) = 1, a(n) = sum_{d | A002110(A055396(n)-1)} moebius(d) * floor(A032742(n) / d).

[Instead of Moebius mu (A008683) one could use Liouville's lambda (A008836) in the above formulas, because all primorials (A002110) are squarefree. A020639(n) gives the smallest prime dividing n, and A055396 gives its index].

a(0) = 0, a(1) = 1, a(2n) = n, a(2n+1) = a(A250470(2n+1)). [After a similar recursive formula for A246277. However, this cannot be used for computing the sequence, unless a definition for A250470(n) is found which doesn't require computing the value of A078898(n).]

For n > 1: a(n) = A249810(n) - A249820(n).

(End)

Other identities:

a(2*n) = n.

For n > 1: a(n)=1 if and only if n is prime.

For n > 1: a(n) = A249808(n, A055396(n)) = A249809(n, A055396(n)).

For n > 1: a(n) = A246277(A249818(n)).

From Antti Karttunen, Jan 04 2015: (Start)

a(n) = 2 if and only if n is a square of a prime.

For all n >= 1: a(A251728(n)) = A243055(A251728(n)) + 2. That is, if n is a semiprime of the form prime(i)*prime(j), prime(i) <= prime(j) < prime(i)^2, then a(n) = (j-i)+2.

(End)

a(A000040(n)^2) = 2; a(A000040(n)*A000040(n+1)) = 3. - Reinhard Zumkeller, Apr 06 2015

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum_{k>=1} (A038110(k)/A038111(k))^2 = 0.2847976823663... . - Amiram Eldar, Oct 26 2024

Maple

N:= 1000: # to get a(0) to a(N)
Primes:= select(isprime, [2, seq(2*i+1, i=1..floor((N-1)/2))]):
A:= Vector(N):
for p in Primes do
  t:= 1:
  A[p]:= 1:
  for n from p^2 to N by p do
    if A[n] = 0 then
       t:= t+1:
       A[n]:= t
    fi
  od
od:
0, 1, seq(A[i], i=2..N); # Robert Israel, Jan 04 2015

Mathematica

Module[{nn=90, spfs}, spfs=Table[FactorInteger[n][[1, 1]], {n, nn}]; Table[ Count[ Take[spfs, i], spfs[[i]]], {i, nn}]] (* Harvey P. Dale, Sep 01 2014 *)

Prog

(PARI)
\\ Not practical for computing, but demonstrates the sum moebius formula:
A020639(n) = { if(1==n, n, vecmin(factor(n)[, 1])); };
A055396(n) = { if(1==n, 0, primepi(A020639(n))); };
A002110(n) = prod(i=1, n, prime(i));
A078898(n) = { my(k, p); if(1==n, n, k = A002110(A055396(n)-1); p = A020639(n); sumdiv(k, d, moebius(d)*(n\(p*d)))); };
\\ Antti Karttunen, Dec 05 2014
(Scheme, with memoizing definec-macro)
(definec (A078898 n) (if (< n 2) n (+ 1 (A078898 (A249744 n)))))
;; Much better for computing. Needs also code from A249738 and A249744. - Antti Karttunen, Dec 06 2014
(Haskell)
import Data.IntMap (empty, findWithDefault, insert)
a078898 n = a078898_list !! n
a078898_list = 0 : 1 : f empty 2 where
   f m x = y : f (insert p y m) (x + 1) where
           y = findWithDefault 0 p m + 1
           p = a020639 x
-- Reinhard Zumkeller, Apr 06 2015

Crossrefs

Cf. A002110, A008683, A008836, A020639, A032742, A054272, A055396, A078899, A078896, A083140, A083221, A243055, A246277, A249738, A249744, A249808, A249809, A249810, A249820, A249818, A250470, A250474, A250477, A250478, A251719, A251724, A251728.

Cf. A001248, A006094, A038110, A038111, A090076.

Sequence in context: A300726 A325352 A305438 * A246277 A260739 A130747

Adjacent sequences: A078895 A078896 A078897 * A078899 A078900 A078901

Keyword

nonn

Author

Reinhard Zumkeller, Dec 12 2002

Extensions

a(0) = 0 prepended for recurrence's sake by Antti Karttunen, Dec 06 2014

Status

approved