A242793 -id:A242793 - 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

A242805 Integers n such that each of n, n + 1, n + 2, n + 4, n + 5, n + 6 is the squarefree product of three primes.

Original entry on oeis.org

73293, 120237, 122613, 130429, 143493, 147953, 171893, 180965, 199833, 213153, 219201, 268017, 287493, 298433, 299553, 300093, 313701, 329793, 332889, 341781, 363597, 369393, 376201, 392509, 404453, 432393, 460801, 475809, 493597, 503457, 506517, 508677

Offset: 1

Daniel Constantin Mayer, May 23 2014

nonn

It is remarkable that this sequence starts with considerably bigger density than the analog A242804 for squarefree integers with two prime divisors. The exceptional density causes the problem that overlapping sextets appear very soon and rather frequently, whereas in A242804 the phenomenon of overlapping sextets does not occur up to the bound 9*10^9.

In fact, there exist 114 nonets n, n + 1, n + 2, n + 4, n + 5, n + 6, n + 8, n + 9, n + 10 of squarefree integers with exactly three prime divisors, up to 10^8. The PARI script in PROG does not start a new sextet before the previous sextet was completed. The impact on bigger clusters, such as nonets and dodekuplets, is illustrated in the CAVEAT of section EXAMPLE.

			73293 = 3*11*2221, 73294 = 2*13*2819, 73295 = 5*107*137,
73297 = 7*37*283,  73298 = 2*67*547,  73299 = 3*53*461.
CAVEAT:
(1) For the dodekuplet, which starts together with the first nonet,
969833 = 17*89*641,  969834 = 2*3*161639, 969835 = 5*31*6257,
969837 = 3*11*29389, 969838 = 2*173*2803, 969839 = 13*61*1223,
969841 = 23*149*283, 969842 = 2*59*8219,  969843 = 3*7*46183,
969845 = 5*47*4127,  969846 = 2*3*161641, 969847 = 29*53*631.
Not all PARI scripts list 969833 and 969841, but not 969837.
(2) For the second nonet,
1450257 = 3*229*2111, 1450258 = 2*179*4051, 1450259 = 83*101*173,
1450261 = 29*43*1163, 1450262 = 2*11*65921, 1450263 = 3*191*2531,
1450265 = 5*23*12611, 1450266 = 2*3*241711, 1450267 = 7*13*15937,
the PARI script lists 1450257 only, but not 1450261.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 176 terms from Zak Seidov)

Cf. A242793 and A242804 (two primes), A242806 (four primes), A242829 (five primes).

s = {}; Do[If[AllTrue[{k, k + 1, k + 2, k + 4, k + 5, k + 6}, SquareFreeQ] && {3, 3, 3, 3, 3, 3} == PrimeOmega[{k, k + 1, k + 2, k + 4, k + 5, k + 6}], AppendTo[s, k]], {k, 73293, 2000000, 4}]; s (* Zak Seidov, Nov 12 2018 *)

{ default(primelimit, 1000M); i=0; j=0; k=0; l=0; m=0; loc=0; lb=2; ub=9*10^9; o=3; 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

is(n) = {my(f=factor(n)); matsize(f)==[3, 2] && vecmax(f[ , 2])==1};
isok(v) = vecextract(v, "^4")==[1, 1, 1, 1, 1, 1]; v = vector(7); for(k=8, 550000, v=concat(vecextract(v, "^1"), is(k+6)); if(isok(v), print1(k, ", "))) \\ Amiram Eldar, Nov 13 2018

upto(n) = {my(res = List(), streak = 1); for(i = 31, n+6, if(factor(i)[, 2] == [1, 1, 1]~, streak++; if(streak % 4 == 3 && streak >= 7, listput(res, i-6)), if(streak % 4 == 3, streak++, streak = 0))); res} \\ David A. Corneth, Nov 13 2018

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

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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.

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A242805 Integers n such that each of n, n + 1, n + 2, n + 4, n + 5, n + 6 is the squarefree product of three primes.

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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.

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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.

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