A120809 - OEIS

%I #14 Aug 05 2024 08:32:19

%S 1859,331169,2141399,4641629,6633419,8447039,10338119,13526009,

%T 20163059,21603425,24099569,26187119,26483321,28226549,33379569,

%U 33485139,40790009,50139819,52046075,56152179,57170075,59824925,72541799,81638579,104151839,106624359,106791269

%N Integers of the form p^2*q in A120806: x+d+1 is prime for all divisors d of x. Both p and q are odd primes, with p and q distinct. See A054753.

%H Amiram Eldar, <a href="/A120809/b120809.txt">Table of n, a(n) for n = 1..1000</a>

%e a(1) = 1859 since x = 11*13^2, divisors(x) = {1, 11, 13, 11*13, 13^2, 11*13^2} and x+d+1 = {1861, 1871, 1873, 2003, 2029, 3719} are all primes.

%p with(numtheory); is3almostprime := proc(n) local L; if n in [0,1] or isprime(n) then return false fi; L:=ifactors(n)[2]; if nops(L) in [1,2,3] and convert(map(z-> z[2], L), `+`) = 3 then return true else return false fi; end; L:=[]: for w to 1 do for k from 1 while nops(L)<=50 do x:=2*k+1; y:=simplify(x^(1/3)); if x mod 6 = 5 and not type(y,integer) #clunky and not issqrfree(x) and is3almostprime(x) and andmap(isprime,[x+2,2*x+1]) then S:=divisors(x); Q:=map(z-> x+z+1, S); if andmap(isprime,Q) then L:=[op(L),x]; print(nops(L),ifactor(x)); fi; fi; od od;

%o (PARI) is(n) = my(f); if(!(n%2), return(0)); f = factor(n); if(f[,2] != [1,2]~ && f[,2] != [2,1]~, return(0)); fordiv(f, d, if(!isprime(n + d + 1), return(0))); 1; \\ _Amiram Eldar_, Aug 05 2024

%Y Intersection of A054753 and A120806.

%K nonn

%O 1,1

%A _Walter Kehowski_, Jul 06 2006

%E a(2) corrected and a(24)-a(27) added by _Amiram Eldar_, Aug 05 2024