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Numbers k that do not have prime anti-divisors.
4

%I #19 Aug 20 2014 20:40:37

%S 1,2,6,30,36,54,90,96,114,120,156,174,210,216,300,330,414,510,516,546,

%T 576,660,714,726,744,804,810,834,894,936,966,1014,1044,1056,1134,1170,

%U 1296,1344,1356,1500,1560,1584,1626,1650,1680,1686,1734,1764,1770,1836,1884,1926,2010,2046,2064

%N Numbers k that do not have prime anti-divisors.

%H 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 a(3) = 6, since 6 has the anti-divisor 4, and it is composite.

%e a(4) = 30, since 30 has the anti-divisors {4, 12, 20} and none are prime.

%e 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 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 (Python)

%o from sympy import isprime, divisors

%o A241557 = [n for n in range(1,10**6) if not any([isprime(x) for x in

%o ..........[2*d for d in divisors(n) if n > 2*d and n % (2*d)] +

%o ..........[d for d in divisors(2*n-1) if n > d >=2 and n % d] +

%o ..........[d for d in divisors(2*n+1) if n > d >=2 and n % d]])]

%o # _Chai Wah Wu_, Aug 19 2014

%Y Cf. A066272, A241556.

%K nonn

%O 1,2

%A _Michael De Vlieger_, Aug 08 2014