login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

LINKS

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

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.

MATHEMATICA

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 *)

PROG

(PARI)

{ 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<n, loc=1; i=n; ); ); if(3==loc, if(j+1==n, k=n; ); if(j+1<n, loc=1; i=n; ); ); if(4==loc, if(k+2==n, l=n; ); if(k+2<n, loc=1; i=n; ); ); if(5==loc, if(l+1==n, m=n; ); if(l+1<n, loc=1; i=n; ); ); if(6==loc, if(m+1==n, print1(i, ", "); loc=0; ); if(m+1<n, loc=1; i=n; ); ); ); ); }

(PARI)

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

(PARI) 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

CROSSREFS

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

Sequence in context: A023185 A244254 A184769 * A250838 A105648 A337740

Adjacent sequences:  A242802 A242803 A242804 * A242806 A242807 A242808

KEYWORD

nonn

AUTHOR

Daniel Constantin Mayer, May 23 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 28 16:37 EDT 2022. Contains 354119 sequences. (Running on oeis4.)