A078898 - OEIS

%I #61 Oct 26 2024 10:45:52

%S 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,

%T 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,

%U 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

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

%C From _Antti Karttunen_, Dec 06 2014: (Start)

%C 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.

%C (End)

%H Harvey P. Dale (terms 1 - 1000) & Antti Karttunen, <a href="/A078898/b078898.txt">Table of n, a(n) for n = 0..10000</a>

%F Ordinal transform of A020639 (Lpf). - _Franklin T. Adams-Watters_, Aug 28 2006

%F From _Antti Karttunen_, Dec 05-08 2014: (Start)

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

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

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

%F [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].

%F 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).]

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

%F (End)

%F Other identities:

%F a(2*n) = n.

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

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

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

%F From _Antti Karttunen_, Jan 04 2015: (Start)

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

%F 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.

%F (End)

%F a(A000040(n)^2) = 2; a(A000040(n)*A000040(n+1)) = 3. - _Reinhard Zumkeller_, Apr 06 2015

%F 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

%p N:= 1000: # to get a(0) to a(N)

%p Primes:= select(isprime, [2,seq(2*i+1,i=1..floor((N-1)/2))]):

%p A:= Vector(N):

%p for p in Primes do

%p   t:= 1:

%p   A[p]:= 1:

%p   for n from p^2 to N by p do

%p     if A[n] = 0 then

%p        t:= t+1:

%p        A[n]:= t

%p     fi

%p   od

%p od:

%p 0,1,seq(A[i],i=2..N); # _Robert Israel_, Jan 04 2015

%t 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 *)

%o (PARI)

%o \\ Not practical for computing, but demonstrates the sum moebius formula:

%o A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };

%o A055396(n) = { if(1==n,0,primepi(A020639(n))); };

%o A002110(n) = prod(i=1, n, prime(i));

%o 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)))); };

%o \\ _Antti Karttunen_, Dec 05 2014

%o (Scheme, with memoizing definec-macro)

%o (definec (A078898 n) (if (< n 2) n (+ 1 (A078898 (A249744 n)))))

%o ;; Much better for computing. Needs also code from A249738 and A249744. - _Antti Karttunen_, Dec 06 2014

%o (Haskell)

%o import Data.IntMap (empty, findWithDefault, insert)

%o a078898 n = a078898_list !! n

%o a078898_list = 0 : 1 : f empty 2 where

%o    f m x = y : f (insert p y m) (x + 1) where

%o            y = findWithDefault 0 p m + 1

%o            p = a020639 x

%o -- _Reinhard Zumkeller_, Apr 06 2015

%Y 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.

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

%K nonn

%O 0,5

%A _Reinhard Zumkeller_, Dec 12 2002

%E a(0) = 0 prepended for recurrence's sake by _Antti Karttunen_, Dec 06 2014