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
Keywords
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
Programs
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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);
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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 *)
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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
Comments