A160394 Numbers n = p*q*r (p, q, r prime) congruent to 0 mod p+q+r.

27, 30, 70, 105, 231, 286, 627, 646, 805, 897, 1581, 1798, 2967, 3055, 3526, 4543, 5487, 6461, 6745, 7198, 7881, 9717, 10366, 10707, 14231, 16377, 20806, 21091, 23326, 26331, 29607, 33901, 35905, 37411, 38086, 38843, 40587, 42211, 44998, 55581

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 12 2009

Keywords

• nonn

Comments

Also numbers n = p*q*r where r = p*q-(p+q) and p, q, r are prime.

For each twin prime pair (q, q+2) the number n = 2*p*(p+2) is in the sequence, since 2+p+(p+2) divides n.

In some cases the factors of n are in arithmetic progression; occurring common differences are 2, 4, 8, 10, 14, 20, 28, 34, 38, 40, 50, 68, 80, 94, 98, ...

All those arithmetic progressions have first term 3, their common differences are the numbers d such that A088420(d/2) = 3. - Klaus Brockhaus, May 17 2009

Examples

			27 = 3*3*3 = (3+3+3)*3, hence 27 is in the sequence; r = 3*3-(3+3).
30 = 2*5*3 = (2+5+3)*3, hence 30 is in the sequence; r = 2*5-(2+5).
165 = 3*5*11 is not a multiple of 3+5+11 = 19, hence 165 is not in the sequence.
627 = 3*11*19 = (3+11+19)*19, hence 627 is in the sequence; r = 3*11-(3+11). The factors 3, 11, 19 are in arithmetic progression (d=8).
40587 = 3*83*163 = (3+83+163)*163, hence 40587 is in the sequence; r = 3*83-(3+83). The factors 3, 83, 163 are in arithmetic progression (d=80).
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A014612 (3-almost primes, numbers that are divisible by exactly 3 primes (counted with multiplicity)).

Cf. A001359 (lesser of twin primes), A115334 (numbers n such that 3+2n and 3+4n are prime), A088420 (number of primes in arithmetic progression starting with 3 and with d=2n). [From Klaus Brockhaus, May 17 2009]

Programs

Magma

[ n: n in [2..56000] | &+[ d[2]: d in f ] eq 3 and n mod &+[ d[1]*d[2]: d in f ] eq 0 where f is Factorization(n) ]; // Klaus Brockhaus, May 17 2009

PARI

list(lim)=my(v=List()); forprime(p=2,lim\4, forprime(q=2,lim\(2*p), my(pq=p*q, r=pq-p-q); if(isprime(r), listput(v, pq*r)))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

Extensions

Partially edited by N. J. A. Sloane, May 14 2009

Missed entry 27 contributed by Zak Seidov, May 14 2009

Further edited by Klaus Brockhaus, May 17 2009