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 A156759 a(1)=2, a(n+1) is the smallest composite number > a(n) with smallest prime factor >= smallest prime factor of a(n). 13
 2, 4, 6, 8, 9, 15, 21, 25, 35, 49, 77, 91, 119, 121, 143, 169, 221, 247, 289, 323, 361, 437, 529, 667, 713, 841, 899, 961, 1147, 1271, 1333, 1369, 1517, 1591, 1681, 1763, 1849, 2021, 2209, 2491, 2773, 2809, 3127, 3233, 3481, 3599, 3721, 4087, 4331, 4453, 4489 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Apart from a(1), this is a sequence of increasing composites such that the derived sequence of their least prime factors is nondecreasing. - R. J. Mathar, Feb 20 2009 Except for a(1)=2, this is the sequence of numbers k such that the smallest prime factor of k is the largest prime less than or equal to the square root of k. - Michael J. Hardy, Nov 29 2013 If, using the standard primality test for a number N by dividing N by consecutive primes <= sqrt(N), it is only on the last step that we conclude that N is not prime, then we call N a "preprime". So, by the last comment, the sequence of preprimes coincides with this sequence for n>=2. Note that, except for 8, all preprimes are semiprimes. - Vladimir Shevelev, Sep 14 2014 LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 FORMULA For n>1, lpf(a(n)) = prime(pi(sqrt(a(n))), where pi(n) = A000720(n). - Vladimir Shevelev, Sep 17 2014 EXAMPLE a(1)=2; a(2)=4=2*2 (2=2) where 2=2; a(3)=6=3*2 (3>2) where 2=2; a(4)=8=2*2*2 (2=2=2) where 2=2; a(5)=9=3*3 (3=3) where 3>2; a(6)=15=5*3 (5>3) where 3=3; a(7)=21=7*3 (7>3) where 3=3; a(8)=25=5*5 (5>3) where 5>3, etc. MAPLE A020639 := proc(n) min(op(numtheory[factorset](n))) ; end: A156759 := proc(n) option remember ; local a; if n = 1 then 2; else for a from procname(n-1)+1 do if not isprime(a) then if A020639(a) >= A020639(procname(n-1)) then RETURN(a) ; fi; fi; od: fi; end: seq(A156759(n), n=1..100) ; # R. J. Mathar, Feb 20 2009 MATHEMATICA lpf[n_] := FactorInteger[n][[1, 1]]; a[1] = 2; a[n_] := a[n] = Module[{k = a[n - 1] + 1, p = lpf[a[n - 1]]}, While[PrimeQ[k] || lpf[k] < p, k++]; k]; Array[a, 100] (* Amiram Eldar, Sep 19 2019 *) nxt[n_]:=Module[{k=n+1, spf}, spf=FactorInteger[n][[1, 1]]; While[PrimeQ[k] || FactorInteger[k][[1, 1]]

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Last modified December 6 10:23 EST 2022. Contains 358630 sequences. (Running on oeis4.)