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A241557 Numbers k that do not have prime anti-divisors.

%I A241557 #21 May 22 2025 10:21:37
%S A241557 1,2,6,30,36,54,90,96,114,120,156,174,210,216,300,330,414,510,516,546,
%T A241557 576,660,714,726,744,804,810,834,894,936,966,1014,1044,1056,1134,1170,
%U A241557 1296,1344,1356,1500,1560,1584,1626,1650,1680,1686,1734,1764,1770,1836,1884,1926,2010,2046,2064
%N A241557 Numbers k that do not have prime anti-divisors.
%H A241557 Chai Wah Wu, <a href="/A241557/b241557.txt">Table of n, a(n) for n = 1..10000</a> (n=1...167 from Michael De Vlieger)
%e A241557 a(3) = 6, since 6 has the anti-divisor 4, and it is composite.
%e A241557 a(4) = 30, since 30 has the anti-divisors {4, 12, 20} and none are prime.
%e A241557 All the integers 6 < k < 30 have at least one prime anti-divisor, and the only integers k < 6 that do not have prime antidivisors are k = {1, 2}.
%t A241557 primeAntiDivisors[n_] := Select[Cases[Range[2, n - 1], _?(Abs[Mod[n, #] - #/2] < 1 &)], PrimeQ]; a241556[n_Integer] := Map[Length[primeAntiDivisors[#]] &, Range[n]]; Flatten[Position[a241556[10^5],0]]
%o A241557 (Python)
%o A241557 from sympy import isprime, divisors
%o A241557 A241557 = [n for n in range(1,10**6) if not any([isprime(x) for x in
%o A241557           [2*d for d in divisors(n) if n > 2*d and n % (2*d)] +
%o A241557           [d for d in divisors(2*n-1) if n > d >=2 and n % d] +
%o A241557           [d for d in divisors(2*n+1) if n > d >=2 and n % d]])]
%o A241557 # _Chai Wah Wu_, Aug 19 2014
%Y A241557 Cf. A066272, A241556.
%K A241557 nonn
%O A241557 1,2
%A A241557 _Michael De Vlieger_, Aug 08 2014