A120810 Integers x of the form p*q*r in A120806: x+d+1 is prime for all divisors d of x, where p, q and r are distinct odd primes. See A007304.
935, 305015, 2339315, 3690185, 14080121, 14259629, 16143005, 17754869, 18679409, 26655761, 29184749, 47372135, 80945699, 82768529, 87102509, 123581021, 131225889, 141328979, 177392861, 224285529, 289174109, 348095519, 350342279, 359093699, 372823805, 403685135
Offset: 1
Keywords
Examples
a(1) = 935 since x = 935 = 5*11*17, divisors(x) = {1, 5, 11, 17, 5*11, 5*17, 11*17, 5*11*17} and x+d+1 = {937, 941, 947, 953, 991, 1021, 1123, 1871} are all primes.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..1000
Programs
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Maple
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; if x mod 6 = 5 and 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;
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PARI
is(n) = my(f); if(!(n%2), return(0)); f = factor(n); if(f[,2] != [1,1,1]~, return(0)); fordiv(f, d, if(!isprime(n + d + 1), return(0))); 1; \\ Amiram Eldar, Aug 05 2024
Extensions
a(16)-a(26) from Amiram Eldar, Aug 05 2024