1,1

a(n) = n-th element of A120806 of the form p*q*r where p, q and r are distinct odd primes.

a(1)=935 since x=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 prime.

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;

Cf. A120806, A007304, A120809, A120808, A120807.

Sequence in context: A171348 A013545 A029570 this_sequence A027554 A031902 A104917

Adjacent sequences: A120807 A120808 A120809 this_sequence A120811 A120812 A120813

nonn

Walter Kehowski (wkehowski(AT)cox.net), Jul 06 2006

1,2

No a(n) can be even, since a(n)+2 must be prime. If a(n) is a prime, then it is a Sophie Germain twin prime (A045536). The only square is 9. Let the degree of n be the sum of the exponents in its prime factorization. By convention, degree(1)=0. Then every a(n) has degree less than or equal to 3. Let the weight of n be the number of its distinct prime factors. By convention, weight(1)=0. Clearly, w<=d is always true, with d=w only when the number is square-free. Let [w,d] be the set of all integers with weight w and degree d. Then only the following possibilities occur: 1. [0,0] => a(1)=1. 2. [1,1] => Sophie Germain twin prime: 3, 5, 11, 29, A005384, A045536. 3. [1,2] => a(4)=9 is the only occurrence. 4. [1,3] => 5^3, 71^3 and 303839^3 are the first few cubes, A000578, A120808. 5. [2,2] => 5*7, 3*13 and 5*13 are the first few semiprimes, A001358, A120807. 6. [2,3] => 11*13^2, 61^2*89 and 13^2*12671 are the first few examples, A014612, A054753, A120809. 7. [3,3] => 5*11*17, 5*53*1151, 5*11*42533 are the first few 3-almost primes, A007304, A120810.

a(n) = n-th number such that n+d+1 is prime for all divisors d of n.

a(11)=125 since divisors(125)={1,5,25,125} and the set of all n+d+1 is {127,131,151,251} and these are all prime.

with(numtheory); L:=[1]: for w to 1 do for k from 1 to 12^6 while nops(L)<=1000 do x:=2*k+1; if andmap(isprime, [x+2, 2*x+1]) then S:=divisors(x) minus {1, 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; L;

Cf. A000578, A001358, A005384, A007304, A014612, A054753, A120776, A120807, A120808, A120809, A120810.

Sequence in context: A092917 A163778 A160358 this_sequence A020946 A091785 A003075

Adjacent sequences: A120803 A120804 A120805 this_sequence A120807 A120808 A120809

nonn

Walter Kehowski (wkehowski(AT)cox.net), Jul 06 2006