A242804 -id:A242804 - cp's OEIS frontend

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.

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

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

A257044 a(n) = (A242804(n)-9)/12.

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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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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.

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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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