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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
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
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]]<spf, k++]; k]; NestList[nxt, 2, 60] (* Harvey P. Dale, Apr 23 2020 *)
CROSSREFS
Sequence in context: A227979 A349151 A080223 * A340609 A340606 A276138
KEYWORD
nonn
AUTHOR
EXTENSIONS
Corrected by R. J. Mathar, Feb 20 2009
STATUS
approved