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Number of distinct prime divisors of n that are <= sqrt(n).
37

%I #32 Mar 07 2021 19:04:44

%S 0,0,0,1,0,1,0,1,1,1,0,2,0,1,1,1,0,2,0,1,1,1,0,2,1,1,1,1,0,3,0,1,1,1,

%T 1,2,0,1,1,2,0,2,0,1,2,1,0,2,1,2,1,1,0,2,1,2,1,1,0,3,0,1,2,1,1,2,0,1,

%U 1,3,0,2,0,1,2,1,1,2,0,2,1,1,0,3,1,1,1,1,0,3,1,1,1,1,1,2,0,2,1,2,0,2,0,1,3

%N Number of distinct prime divisors of n that are <= sqrt(n).

%C For all primes p: a(p) = 0 (not marked) and for k > 1 a(p^k) = 1.

%C a(1) = 0 and for n > 0 a(n) is the number of marks when applying the sieve of Eratosthenes where a stage for prime p starts at p^2.

%C If we define a divisor d|n to be inferior if d <= n/d, then inferior divisors are counted by A038548 and listed by A161906. This sequence counts inferior prime divisors. - _Gus Wiseman_, Feb 25 2021

%H Harry J. Smith, <a href="/A063962/b063962.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: Sum_{k>=1} x^(prime(k)^2) / (1 - x^prime(k)). - _Ilya Gutkovskiy_, Apr 04 2020

%F a(A002110(n)) = n for n > 2. - _Gus Wiseman_, Feb 25 2021

%e a(33) = a(3*11) = 1, as 3^2 = 9 < 33 and 11^2 = 121 > 33.

%e From _Gus Wiseman_, Feb 25 2021: (Start)

%e The a(n) inferior prime divisors (columns) for selected n:

%e n = 3 8 24 3660 390 3570 87780

%e ---------------------------------

%e {} 2 2 2 2 2 2

%e 3 3 3 3 3

%e 5 5 5 5

%e 13 7 7

%e 17 11

%e 19

%e (End)

%p with(numtheory): a:=proc(n) local c,F,f,i: c:=0: F:=factorset(n): f:=nops(F): for i from 1 to f do if F[i]^2<=n then c:=c+1 else c:=c: fi od: c; end: seq(a(n),n=1..105); # _Emeric Deutsch_

%t Join[{0},Table[Count[Transpose[FactorInteger[n]][[1]],_?(#<=Sqrt[n]&)],{n,2,110}]] (* _Harvey P. Dale_, Mar 26 2015 *)

%o (PARI) { for (n=1, 1000, f=factor(n)~; a=0; for (i=1, length(f), if (f[1, i]^2<=n, a++, break)); write("b063962.txt", n, " ", a) ) } \\ _Harry J. Smith_, Sep 04 2009

%o (Haskell)

%o a063962 n = length [p | p <- a027748_row n, p ^ 2 <= n]

%o -- _Reinhard Zumkeller_, Apr 05 2012

%Y Cf. A055399, A001221.

%Y Cf. A027748, A063962.

%Y Zeros are at indices A008578.

%Y The divisors are listed by A161906 and add up to A097974.

%Y Dominates A333806 (the strictly inferior version).

%Y The superior version is A341591.

%Y The strictly superior version is A341642.

%Y A001221 counts prime divisors, with sum A001414.

%Y A033677 selects the smallest superior divisor.

%Y A038548 counts inferior divisors.

%Y A063538/A063539 have/lack a superior prime divisor.

%Y A161908 lists superior divisors.

%Y A207375 lists central divisors.

%Y A217581 selects the greatest inferior prime divisor.

%Y A341676 lists the unique superior prime divisors.

%Y - Inferior: A033676, A066839, A069288, A072499, A333749, A333750.

%Y - Superior: A051283, A059172, A070038, A072500, A116883, A341592, A341675.

%Y - Strictly Inferior: A056924, A060775, A070039, A333805, A341596, A341674.

%Y - Strictly Superior: A056924, A140271, A238535, A341594, A341595, A341673.

%Y Cf. A000005, A001248, A005117, A006530, A020639, A048098, A064052, A341643.

%K nonn

%O 1,12

%A _Reinhard Zumkeller_, Sep 04 2001

%E Revised definition from _Emeric Deutsch_, Jan 31 2006