A242804 Integers k such that each of k, k+1, k+2, k+4, k+5, k+6 is the product of two distinct primes.

213, 143097, 194757, 206133, 273417, 684897, 807657, 1373937, 1391757, 1516533, 1591593, 1610997, 1774797, 1882977, 1891761, 2046453, 2051493, 2163417, 2163957, 2338053, 2359977, 2522517, 2913837, 3108201, 4221753

Offset: 1

Author

Daniel Constantin Mayer, May 23 2014

Keywords

• nonn

Comments

A remarkable gap occurs between the initial two members, and the sequence seems to be rather sparse compared to the related A242805.

Here, the first member k of the sextet is the reference, whereas in A068088 the center k+3 is selected as reference. Observe that k+3 must be divisible by the square 4.

All terms are congruent to 9 (mod 12). - Zak Seidov, Apr 14 2015

From Robert Israel, Apr 15 2015: (Start)

All terms are congruent to 33 (mod 36).

Numbers k in A039833 such that k+4 is in A039833. (End)

From Robert G. Wilson v, Apr 15 2015: (Start)

k is congruent to 33 (mod 36) so one of its factors is 3 and the other is == 11 (mod 12);

k+1 is congruent to 34 (mod 36) so one of its factors is 2 and the other is == 17 (mod 18);

k+2 is congruent to 35 (mod 36) so its factors are == +-1 (mod 6);

k+4 is congruent to 1 (mod 36) so its factors are == +-1 (mod 6);

k+5 is congruent to 2 (mod 36) so one of its factors is 2 and the other is == 1 (mod 18);

k+6 is congruent to 3 (mod 36) so one of its factors is 3 and the other is == 1 (mod 12). (End).

Number of terms < 10^m: 0, 0, 1, 1, 1, 7, 39, 169, 882, 4852, 27479, ...,. - Robert G. Wilson v, Apr 15 2015

Or, numbers k such that k, k+1 and k+2 are terms in A175648. - Zak Seidov, Dec 08 2015

Examples

			213=3*71, 214=2*107, 215=5*43, 217=7*31, 218=2*109, 219=3*73.
		

Links

• Zak Seidov and Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A242793 (minima for two, three and more prime divisors) and A068088 (arbitrary squarefree integers).

Cf. A242805, A039833, A175648, A202319.

Programs

Maple

f:= t -> numtheory:-issqrfree(t) and (numtheory:-bigomega(t) = 2):
select(t -> andmap(f, [t,t+1,t+2,t+4,t+5,t+6]), [seq(36*k+33,k=0..10^6)]); # Robert Israel, Apr 15 2015

Mathematica

fQ[n_] := PrimeQ[n/3] && PrimeQ[(n + 1)/2] && PrimeQ[(n + 5)/2] && PrimeQ[(n + 6)/3] && PrimeNu[{n + 2, n + 4}] == {2, 2} == PrimeOmega[{n + 2, n + 4}]; k = 33; lst = {}; While[k < 10^8, If[fQ@ k, AppendTo[lst, k]]; k += 36]; lst (* Robert G. Wilson v, Apr 14 2015 and revised Apr 15 2015 after Zak Seidov and Robert Israel *)

PARI

default(primelimit, 1000M); i=0; j=0; k=0; l=0; m=0; loc=0; lb=2; ub=9*10^9; o=2; for(n=lb, ub, if(issquarefree(n)&&(o==omega(n)), loc=loc+1; if(1==loc, i=n; ); if(2==loc, if(i+1==n, j=n; ); if(i+1
				

PARI

forstep(x=213,4221753,12, if( isprime(x/3) && isprime((x+1)/2) && 2==omega(x+2) && 2==bigomega(x+2) && 2==omega(x+4) && 2==bigomega(x+4) && isprime((x+5)/2) && isprime((x+6)/3), print1(x", "))) \\ Zak Seidov, Apr 14 2015

Formula

a(n) = A202319(n) - 1. - Jon Maiga, Jul 10 2021