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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. 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

Zak Seidov, Table of n, a(n) for n = 1..176

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,

the PARI script lists 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

CROSSREFS

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

Sequence in context: A023185 A244254 A184769 * A250838 A105648 A180300

Adjacent sequences:  A242802 A242803 A242804 * A242806 A242807 A242808

KEYWORD

nonn,changed

AUTHOR

Daniel Constantin Mayer, May 23 2014

STATUS

approved

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Last modified November 19 08:48 EST 2018. Contains 317347 sequences. (Running on oeis4.)