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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).

%I A156759 #52 Apr 09 2025 04:21:26
%S A156759 2,4,6,8,9,15,21,25,35,49,77,91,119,121,143,169,221,247,289,323,361,
%T A156759 437,529,667,713,841,899,961,1147,1271,1333,1369,1517,1591,1681,1763,
%U A156759 1849,2021,2209,2491,2773,2809,3127,3233,3481,3599,3721,4087,4331,4453,4489
%N A156759 a(1)=2, a(n+1) is the smallest composite number > a(n) with smallest prime factor >= smallest prime factor of a(n).
%C A156759 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
%C A156759 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
%C A156759 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
%H A156759 Amiram Eldar, <a href="/A156759/b156759.txt">Table of n, a(n) for n = 1..10000</a>
%H A156759 Vladimir Shevelev, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2014-September/013643.html">A classification of the positive integers over primes</a>
%F A156759 For n>1, lpf(a(n)) = prime(pi(sqrt(a(n)))), where pi(n) = A000720(n). - _Vladimir Shevelev_, Sep 17 2014
%e A156759 a(1)=2;
%e A156759 a(2)=4=2*2 (2=2) where 2=2;
%e A156759 a(3)=6=3*2 (3>2) where 2=2;
%e A156759 a(4)=8=2*2*2 (2=2=2) where 2=2;
%e A156759 a(5)=9=3*3 (3=3) where 3>2;
%e A156759 a(6)=15=5*3 (5>3) where 3=3;
%e A156759 a(7)=21=7*3 (7>3) where 3=3;
%e A156759 a(8)=25=5*5 (5>3) where 5>3, etc.
%p A156759 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
%t A156759 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 *)
%t A156759 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 *)
%Y A156759 Cf. A000720 (pi), A002808 (composites), A020639 (lpf), A156604.
%K A156759 nonn
%O A156759 1,1
%A A156759 _Juri-Stepan Gerasimov_, Feb 15 2009
%E A156759 Corrected by _R. J. Mathar_, Feb 20 2009