A156759 a(1)=2, a(n+1) is the smallest composite number > a(n) with smallest prime factor >= smallest prime factor of a(n).

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

Author

Juri-Stepan Gerasimov, Feb 15 2009

Keywords

• nonn

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

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Vladimir Shevelev, A classification of the positive integers over primes

Crossrefs

Cf. A000720 (pi), A002808 (composites), A020639 (lpf), A156604.

Programs

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]]Harvey P. Dale, Apr 23 2020 *)

Formula

For n>1, lpf(a(n)) = prime(pi(sqrt(a(n)))), where pi(n) = A000720(n). - Vladimir Shevelev, Sep 17 2014

Extensions

Corrected by R. J. Mathar, Feb 20 2009