A242966 Composite numbers whose anti-divisors are all primes.

4, 8, 16, 64, 1024, 4096, 65536, 262144, 4194304, 1073741824, 1152921504606846976, 1267650600228229401496703205376, 85070591730234615865843651857942052864, 93536104789177786765035829293842113257979682750464

Offset: 1

Author

Paolo P. Lava, May 28 2014

Keywords

• nonn

Comments

It appears they are all powers of 2.

Subset of A242965.

a(n) must be 2^k. - Hiroaki Yamanouchi, Mar 17 2015

The exponents are 2, 3, 4, 6, 10, 12, 16, 18, 22, 30, 60, 100, 126, 166, 198, ... - Michel Marcus, Mar 18 2015

Examples

			The anti-divisors of 1024 are all primes: 3, 23, 89, 683.
The same for 65536: 3, 43691.
		

Links

• Hiroaki Yamanouchi, Table of n, a(n) for n = 1..15

Crossrefs

Cf. A066272, A242965.

Programs

Maple

P := proc(q) local k,ok,n; for n from 3 to q do if not isprime(n)
then ok:=1; for k from 2 to n-1 do if abs((n mod k)-k/2)<1
then if not isprime(k) then ok:=0; break; fi; fi; od;
if ok=1 then print(n); fi; fi; od; end: P(10^100);

Mathematica

antiDivisors[n_] := Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]; Select[2^Range[2, 20], AllTrue[antiDivisors@ #, PrimeQ] &] (* _Michael De Vlieger, Mar 18 2015 *)

Python

from sympy import isprime, divisors
A242966 = [n for n in range(3,10**5) if not isprime(n) and list(filter(lambda x: not isprime(x), [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 13 2014

Extensions

a(11)-a(14) from Hiroaki Yamanouchi, Mar 17 2015