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

1, 2, 6, 30, 36, 54, 90, 96, 114, 120, 156, 174, 210, 216, 300, 330, 414, 510, 516, 546, 576, 660, 714, 726, 744, 804, 810, 834, 894, 936, 966, 1014, 1044, 1056, 1134, 1170, 1296, 1344, 1356, 1500, 1560, 1584, 1626, 1650, 1680, 1686, 1734, 1764, 1770, 1836, 1884, 1926, 2010, 2046, 2064

Offset: 1

Author

Michael De Vlieger, Aug 08 2014

Keywords

• nonn

Examples

			a(3) = 6, since 6 has the anti-divisor 4, and it is composite.
a(4) = 30, since 30 has the anti-divisors {4, 12, 20} and none are prime.
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}.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n=1...167 from Michael De Vlieger)

Crossrefs

Cf. A066272, A241556.

Programs

Mathematica

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]]

Python

from sympy import isprime, divisors
A241557 = [n for n in range(1,10**6) if not any([isprime(x) for x in
          [2*d for d in divisors(n) if n > 2*d and n % (2*d)] +
          [d for d in divisors(2*n-1) if n > d >=2 and n % d] +
          [d for d in divisors(2*n+1) if n > d >=2 and n % d]])]
# Chai Wah Wu, Aug 19 2014