A242966 Composite numbers whose anti-divisors are all primes.

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

Offset

1,1

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

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A066272, A242965.

Sequence in context: A038238 A230112 A023376 * A337235 A038110 A241197

Adjacent sequences: A242963 A242964 A242965 * A242967 A242968 A242969

Keyword

nonn

Author

Paolo P. Lava, May 28 2014

Extensions

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

Status

approved