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A063962
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Number of distinct prime divisors of n that are <= sqrt(n).
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37
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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, 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, 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
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OFFSET
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1,12
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COMMENTS
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For all primes p: a(p) = 0 (not marked) and for k > 1 a(p^k) = 1.
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.
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
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} x^(prime(k)^2) / (1 - x^prime(k)). - Ilya Gutkovskiy, Apr 04 2020
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EXAMPLE
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a(33) = a(3*11) = 1, as 3^2 = 9 < 33 and 11^2 = 121 > 33.
The a(n) inferior prime divisors (columns) for selected n:
n = 3 8 24 3660 390 3570 87780
---------------------------------
{} 2 2 2 2 2 2
3 3 3 3 3
5 5 5 5
13 7 7
17 11
19
(End)
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MAPLE
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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
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MATHEMATICA
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Join[{0}, Table[Count[Transpose[FactorInteger[n]][[1]], _?(#<=Sqrt[n]&)], {n, 2, 110}]] (* Harvey P. Dale, Mar 26 2015 *)
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PROG
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(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
(Haskell)
a063962 n = length [p | p <- a027748_row n, p ^ 2 <= n]
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CROSSREFS
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Dominates A333806 (the strictly inferior version).
The strictly superior version is A341642.
A033677 selects the smallest superior divisor.
A217581 selects the greatest inferior prime divisor.
A341676 lists the unique superior prime divisors.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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