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A242965 Numbers whose anti-divisors are all primes.

%I A242965 #20 Dec 12 2024 02:52:00
%S A242965 3,4,5,7,8,11,16,17,19,29,43,47,61,64,71,79,89,101,107,109,151,191,
%T A242965 197,223,251,271,317,349,359,421,439,461,521,569,601,631,659,673,691,
%U A242965 701,719,811,821,881,911,919,947,971,991,1009,1024,1051,1091,1109,1153
%N A242965 Numbers whose anti-divisors are all primes.
%H A242965 Paolo P. Lava, <a href="/A242965/b242965.txt">Table of n, a(n) for n = 1..1000</a>
%e A242965 The anti-divisors of 191 are all primes: 2, 3, 127.
%e A242965 The same for 1024: 3, 23, 89, 683.
%p A242965 P := proc(q) local k,ok,n; for n from 3 to q do ok:=1;
%p A242965 for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then
%p A242965 if not isprime(k) then ok:=0; break; fi; fi; od;
%p A242965 if ok=1 then print(n); fi; od; end: P(10^3);
%o A242965 (Python)
%o A242965 from sympy import divisors, isprime
%o A242965 for n in range(3, 10**4):
%o A242965     for d in [2*d for d in divisors(n) if n > 2*d and n % (2*d)] + \
%o A242965              [d for d in divisors(2*n-1) if n > d >= 2 and n % d] + \
%o A242965              [d for d in divisors(2*n+1) if n > d >= 2 and n % d]:
%o A242965         if not isprime(d):
%o A242965             break
%o A242965     else:
%o A242965         print(n, end=', ')
%o A242965 # _Chai Wah Wu_, Aug 15 2014
%Y A242965 Cf. A066272, A242966.
%K A242965 nonn,easy
%O A242965 1,1
%A A242965 _Paolo P. Lava_, May 28 2014