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A081062 Neither 3-smooth numbers nor prime powers.

%I A081062 #21 Sep 16 2024 12:48:02
%S A081062 10,14,15,20,21,22,26,28,30,33,34,35,38,39,40,42,44,45,46,50,51,52,55,
%T A081062 56,57,58,60,62,63,65,66,68,69,70,74,75,76,77,78,80,82,84,85,86,87,88,
%U A081062 90,91,92,93,94,95,98,99,100,102,104,105,106,110,111,112,114,115,116
%N A081062 Neither 3-smooth numbers nor prime powers.
%C A081062 A081060(m) > 1 iff m = a(k) for some k. - corrected by _Gionata Neri_, Jul 30 2016
%C A081062 Complement of A081061.
%C A081062 Composites with smallest prime factor^largest prime factor > largest prime factor^smallest prime factor. - _Juri-Stepan Gerasimov_, Jan 04 2009
%H A081062 Robert Israel, <a href="/A081062/b081062.txt">Table of n, a(n) for n = 1..10000</a>
%e A081062 12 = 2^2*3 is not in the sequence because it is 3-smooth (all prime factors are 3 or less). 17 = 17^1 and 49 = 7^2 are not in the sequence because they are prime powers. - _Michael B. Porter_, Jul 31 2016
%p A081062 filter:= proc(n) local f; f:= numtheory:-factorset(n); nops(f) > 1 and max(f) > 3 end proc:
%p A081062 select(filter, [$1..1000]); # _Robert Israel_, Jul 31 2016
%t A081062 Select[Range@ 120, Nor[PrimePowerQ@ #, 3 EulerPhi[6 #] == 6 #] &] (* _Michael De Vlieger_, Aug 02 2016, after _Robert G. Wilson v_ at A003586 *)
%o A081062 (Python)
%o A081062 from sympy import integer_log, primepi, integer_nthroot
%o A081062 def A081062(n):
%o A081062     def f(x): return int(n+1-(a:=x.bit_length())-(b:=integer_log(x,3)[0])+sum((x//3**i).bit_length() for i in range(b+1))+sum(primepi(integer_nthroot(x, k)[0]) for k in range(1, a)))
%o A081062     m, k = n, f(n)
%o A081062     while m != k: m, k = k, f(k)
%o A081062     return m # _Chai Wah Wu_, Sep 16 2024
%Y A081062 Cf. A003586, A000961.
%K A081062 nonn,easy
%O A081062 1,1
%A A081062 _Reinhard Zumkeller_, Mar 04 2003