A120809 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.

1859, 331169, 2141399, 4641629, 6633419, 8447039, 10338119, 13526009, 20163059, 21603425, 24099569, 26187119, 26483321, 28226549, 33379569, 33485139, 40790009, 50139819, 52046075, 56152179, 57170075, 59824925, 72541799, 81638579, 104151839, 106624359, 106791269

Offset

1,1

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

Example

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.

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; 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;

Prog

(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

Crossrefs

Intersection of A054753 and A120806.

Sequence in context: A064978 A202913 A159212 * A213868 A022062 A107526

Adjacent sequences: A120806 A120807 A120808 * A120810 A120811 A120812

Keyword

nonn

Author

Walter Kehowski, Jul 06 2006

Extensions

a(2) corrected and a(24)-a(27) added by Amiram Eldar, Aug 05 2024

Status

approved