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]]<spf, k++]; k]; NestList[nxt, 2, 60] (* Harvey P. Dale, Apr 23 2020 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Juri-Stepan Gerasimov, Feb 15 2009
EXTENSIONS
Corrected by R. J. Mathar, Feb 20 2009
STATUS
approved