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A066509 a(n) is the first of a triple of consecutive integers, each the product of three distinct primes. 13
1309, 1885, 2013, 2665, 3729, 5133, 6061, 6213, 6305, 6477, 6853, 6985, 7257, 7953, 8393, 8533, 8785, 9213, 9453, 9821, 9877, 10281, 10945, 11605, 12453, 12565, 12801, 12857, 12993, 13053, 14133, 14313, 14329, 14465, 14817, 15085, 15265, 15805, 16113, 16133 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A subsequence of A052214 and thus of A005238. - M. F. Hasler, Jan 05 2013
Also, the start of pairs of adjacent sphenic twins, i.e., a(n) = A215217(k) such that A215217(k+1) = A215217(k)+1. Therefore these triples might be called "sphenic triples". They form a subsequence of A242606. - M. F. Hasler, May 18 2014
Minimal difference is 4 which occurs at indices n = {316, 547, 566, 604, 666, 695, 821, 874, 979, ...}. - Zak Seidov, Jul 04 2020
LINKS
G. L. Honaker, Jr. and C. Caldwell, Prime Curios! 1309
FORMULA
a(n) == 1 (mod 4). - Zak Seidov, Mar 31 2020
EXAMPLE
a(5) = 3729 because it along with 3730 and 3731 are all the product of three distinct primes.
MATHEMATICA
f[n_]:=Last/@FactorInteger[n]=={1, 1, 1}; lst={}; Do[If[f[n]&&f[n+1]&&f[n+2], AppendTo[lst, n]], {n, 9!}]; lst (* Vladimir Joseph Stephan Orlovsky, Mar 04 2010 *)
PROG
(PARI) Trip(n) = { local(f); f=factor(n); if (matsize(f)[1] != 3, return(0)); for(i=1, 3, if (f[i, 2] != 1, return(0))); return(1); } { n=0; for (m=1, 10^10, if (!Trip(m) || !Trip(m+1) || !Trip(m+2), next); write("b066509.txt", n++, " ", m); if (n==1000, return) ) } \\ Harry J. Smith, Feb 19 2010
(PARI) A066509(n, show_all=0, a=2*3*5, s=[1, 1, 1]~)={until( !n-- || !a++, until(, factor(a+2)[, 2]!=s && (a+=3) && next; factor(a+1)[, 2]!=s && (a+=2) && next; factor(a)[, 2]==s && break; factor(a+3)[, 2]==s && a++ && break; a+=4); show_all && print1(a", ")); a} \\ M. F. Hasler, Jan 05 2013
(PARI) is3dp(n)=my(f=factor(n)); matsize(f)==[3, 2]&&vecmax(f[, 2])==1
list(lim)=my(v=List(), t); forprime(p=17, lim\15, forprime(q=7, min(p-1, lim\3), forprime(r=3, min(q-1, lim\(p*q)), t=p*q*r; if(t%4==1 && is3dp(t+1) && is3dp(t+2), listput(v, t))))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jan 05 2013
(PARI) list(lim)=my(v=List(), ct); forfactored(n=1309, lim\1+2, if(n[2][, 2]==[1, 1, 1]~, if(ct++==3, listput(v, n[1]-2)), ct=0)); Vec(v) \\ Charles R Greathouse IV, Aug 30 2022
CROSSREFS
Subsequence of A052214 and hence of A005238.
Subsequence of A215217, A007675, A242606 and A168626.
Sequence in context: A209853 A165936 A242606 * A248202 A256668 A205301
KEYWORD
nonn
AUTHOR
Jason Earls, Jan 04 2002
STATUS
approved

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Last modified May 29 11:28 EDT 2024. Contains 372940 sequences. (Running on oeis4.)