A093549
a(n) is the smallest number m such that each of the numbers m-1, m and m+1 has n distinct prime divisors.
Original entry on oeis.org
3, 21, 645, 37961, 1042405, 323567035, 30989984675, 10042712381261
Offset: 1
a(3)=645 because 644=2^2*7*23; 645=3*5*43; 646=2*17*19 and 645 is the smallest number m such that each of the numbers m-1, m and m+1 has 3 distinct prime divisors.
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a[n_] := (For[m=2, !(Length[FactorInteger[m-1]]==n && Length[FactorInteger[m]]==n&&Length[FactorInteger[m+1]]==n), m++ ];m);Do[Print[a[n]], {n, 7}]
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a(n,m=2)=until(,for(k=-1,1,omega(m-k)!=n&&(m+=2-k)&&next(2));return(m)) \\ M. F. Hasler, May 20 2014
A242492
For any integer m > 1, the m-th term of the sequence is the minimal squarefree integer x with exactly m prime divisors such that x+1 and x+2 are also squarefree integers with exactly m prime divisors.
Original entry on oeis.org
33, 1309, 203433, 16467033, 1990586013, 41704979953, 102099792179229
Offset: 2
33 = 3*11, 34 = 2*17, 35 = 5*7;
1309 = 7*11*17, 1310 = 2*5*131, 1311 = 3*19*23;
203433 = 3*19*43*83, 203434 = 2*7*11*1321, 203435 = 5*23*29*61;
16467033 = 3*11*17*149*197, 16467034 = 2*19*23*83*227, 16467035 = 5*13*37*41*167; (CPU time 48 seconds)
1990586013 = 3*13*29*67*109*241, 1990586014 = 2*23*37*43*59*461, 1990586015 = 5*11*17*19*89*1259. (CPU time 2 hours and 34 minutes)
- Hugh L. Montgomery and Robert C. Vaughan: "Multiplicative Number Theory: 1. Classical Theory", Cambridge studies in advanced mathematics, vol. 97, Cambridge University Press (2007)
Cf.
A242605-
A242608 for start of triples of consecutive squarefree numbers with m=2,...,5 prime factors,
A242621 for the analog of the present sequence in that spirit.
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{default(primelimit,2M); lb=2; ub=2*10^9; m=1; i=0; j=0; loc=0; while(m<6, m=m+1; for(n=lb,ub, if(issquarefree(n)&&(m==omega(n)), loc=loc+1; if(1==loc, i=n; ); if(2==loc, if(i+1==n, j=n; ); if(i+1
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