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Numbers whose anti-divisors are all primes.
3

%I #20 Dec 12 2024 02:52:00

%S 3,4,5,7,8,11,16,17,19,29,43,47,61,64,71,79,89,101,107,109,151,191,

%T 197,223,251,271,317,349,359,421,439,461,521,569,601,631,659,673,691,

%U 701,719,811,821,881,911,919,947,971,991,1009,1024,1051,1091,1109,1153

%N Numbers whose anti-divisors are all primes.

%H Paolo P. Lava, <a href="/A242965/b242965.txt">Table of n, a(n) for n = 1..1000</a>

%e The anti-divisors of 191 are all primes: 2, 3, 127.

%e The same for 1024: 3, 23, 89, 683.

%p P := proc(q) local k,ok,n; for n from 3 to q do ok:=1;

%p for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then

%p if not isprime(k) then ok:=0; break; fi; fi; od;

%p if ok=1 then print(n); fi; od; end: P(10^3);

%o (Python)

%o from sympy import divisors, isprime

%o for n in range(3, 10**4):

%o for d in [2*d for d in divisors(n) if n > 2*d and n % (2*d)] + \

%o [d for d in divisors(2*n-1) if n > d >= 2 and n % d] + \

%o [d for d in divisors(2*n+1) if n > d >= 2 and n % d]:

%o if not isprime(d):

%o break

%o else:

%o print(n, end=', ')

%o # _Chai Wah Wu_, Aug 15 2014

%Y Cf. A066272, A242966.

%K nonn,easy

%O 1,1

%A _Paolo P. Lava_, May 28 2014