A242805 -id:A242805 - cp's OEIS frontend

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

Daniel Constantin Mayer, May 23 2014

nonn

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

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

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

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

Cf. A242805, A039833, A175648, A202319.

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

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 *)

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

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

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

A242793 The minimal integer x such that each of the six integers x, x+1, x+2, x+4, x+5, x+6 is squarefree with exactly n prime divisors.

Original entry on oeis.org

213, 73293, 9743613, 6639266409

Offset: 2

Daniel Constantin Mayer, May 22 2014

This is the next step in my project to study the distribution of increasingly extensive clusters of squarefree integers with fixed number of prime divisors: triples x,x+1,x+2 were investigated in A242492 and here we study sextets x,x+1,x+2,x+4,x+5,x+6 with a central gap x+3, since x+3 must be divisible by the square 4.

The term 6639266409 required 30 hours of CPU time on an iMac with Intel i7 Quadcore CPU running OS X Lion.

			213=3*71, 214=2*107, 215=5*43, 217=7*31, 218=2*109, 219=3*73;
73293=3*11*2221, 73294=2*13*2819, 73295=5*107*137,
73297=7*37*283, 73298=2*67*547, 73299=3*53*461;
9743613=3*11*503*587, 9743614=2*59*71*1163, 9743615= 5*7*167*1667, 9743617=13*37*47*431, 9743618=2*17*19*15083, 9743619=3*83*109*359;
6639266409=3*29*109*421*1663, 6639266410=2*5*7*113*839351,
6639266411=17*23*89*101*1889, 6639266413=13*61*79*131*809,
6639266414=2*11*349*857*1009, 6639266415=3*5*73*149*40693.

Cf. A242492 (triples) and A242804, A242805, A242806, A242829, and also A068088.

{ default(primelimit,1000M); i=0; j=0; k=0; l=0; m=0; loc=0; lb=2; ub=9*10^9; o=1; while(o<5, o=o+1; 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

A242829 Integers n such that each of n, n+1, n+2, n+4, n+5, n+6 is the squarefree product of five primes.

Original entry on oeis.org

6639266409, 8628052209, 12692281897, 14492398389, 15798643881, 18883291565, 20404935965, 20825703713, 21342970293, 21597222381, 22221458853, 22567169229, 22578915665, 23000623161, 23198162685, 23247729109, 24163642653, 24802386189, 24894100941, 26297281109

Offset: 1

Daniel Constantin Mayer, May 23 2014

nonn

This is a higher analog to A242804, A242805, A242806.

It is very tough to compute this sequence on a single machine. Therefore, the interval from 0 to 9*10^9 was subdivided into 9 partial intervals, of length 10^9 each, and scanned by different computers. Nevertheless the CPU time was extrapolated for a single machine and summed up to 30 hours for the first member 6639266409 and 47 hours (~2 days) for the second member 8628052209. Up to 10^10, there is no occurrence of a next term.

			a(1) = 6639266409 =  3 * 29 * 109 * 421 *   1663,
       6639266410 =  2 *  5 *   7 * 113 * 839351,
       6639266411 = 17 * 23 *  89 * 101 *   1889,
       6639266413 = 13 * 61 *  79 * 131 *    809,
       6639266414 =  2 * 11 * 349 * 857 *   1009,
       6639266415 =  3 *  5 *  73 * 149 *  40693;
and
a(2) = 8628052209 =  3 *  7 *  19 * 863 *  25057,
       8628052210 =  2 *  5 * 251 * 953 *   3607,
       8628052211 = 17 * 43 * 179 * 233 *    283,
       8628052213 = 11 * 47 * 127 * 331 *    397,
       8628052214 =  2 * 53 * 107 * 131 *   5807,
       8628052215 =  3 *  5 *  23 *  31 * 806737.

Jens Kruse Andersen, Table of n, a(n) for n = 1..195

Cf. A242793 and A242804, A242805, A242806.

{ default(primelimit, 1000M); i=0; j=0; k=0; l=0; m=0; loc=0; lb=2; ub=9*10^9; o=5; 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

More terms from Jens Kruse Andersen, Jun 18 2014

A242806 Integers k such that each of k, k+1, k+2, k+4, k+5, k+6 is the squarefree product of four primes.

Original entry on oeis.org

9743613, 9780589, 22669381, 23209809, 23395137, 26143197, 28042877, 38110029, 43335609, 45127905, 45559613, 47163489, 47944865, 48030229, 48611913, 48762829, 50015301, 50600341, 50742501, 51256541, 52691613

Offset: 1

Daniel Constantin Mayer, May 23 2014

This is a higher analog to A242804 and A242805.

			9743613=3*11*503*587, 9743614=2*59*71*1163, 9743615= 5*7*167*1667, 9743617=13*37*47*431, 9743618=2*17*19*15083, 9743619=3*83*109*359.

Cf. A242793 and A242804 (two primes), A242805 (three primes), A242829 (five primes).

sfp4Q[n_]:=With[{c=n+{0,1,2,4,5,6}},Union[PrimeNu[c]]=={4}&&AllTrue[c,SquareFreeQ]]; Select[Range[52700000],sfp4Q] (* Harvey P. Dale, Jul 11 2025 *)

{ default(primelimit, 1000M); i=0; j=0; k=0; l=0; m=0; loc=0; lb=2; ub=9*10^9; o=4; 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

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A242793 The minimal integer x such that each of the six integers x, x+1, x+2, x+4, x+5, x+6 is squarefree with exactly n prime divisors.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A242829 Integers n such that each of n, n+1, n+2, n+4, n+5, n+6 is the squarefree product of five primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A242806 Integers k such that each of k, k+1, k+2, k+4, k+5, k+6 is the squarefree product of four primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI