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A066509
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a(n) is the first of a triple of consecutive integers, each the product of three distinct primes.
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13
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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a(5) = 3729 because it along with 3730 and 3731 are all the product of three distinct primes.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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