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A097889 Numbers that are products of (at least two) consecutive primes.

%I A097889 #44 Aug 30 2024 02:55:27
%S A097889 6,15,30,35,77,105,143,210,221,323,385,437,667,899,1001,1147,1155,
%T A097889 1517,1763,2021,2310,2431,2491,3127,3599,4087,4199,4757,5005,5183,
%U A097889 5767,6557,7387,7429,8633,9797,10403,11021,11663,12317,12673,14351,15015,16637,17017
%N A097889 Numbers that are products of (at least two) consecutive primes.
%C A097889 Subsequence of A073485; A073490(a(n)) = 0. - _Reinhard Zumkeller_, Nov 20 2004
%C A097889 A proper subset of A073485. - _Robert G. Wilson v_, Jun 11 2010
%C A097889 A192280(a(n)) * (1 - A010051(a(n))) = 1. - _Reinhard Zumkeller_, Aug 26 2011 [corrected by _Jason Yuen_, Aug 29 2024]
%C A097889 The Heinz numbers of the partitions into at least 2 consecutive parts. The Heinz number of an integer partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by _Alois P. Heinz_ in A215366 as an "encoding" of a partition). Examples: (i) 105 (=3*5*7) is in the sequence because it is the Heinz number of the partition [2,3,4]; (ii) 108 (= 2*2*3*3*3) is not in the sequence because it is the Heinz number of the partition [1,1,2,2,2]. - _Emeric Deutsch_, Oct 02 2015
%H A097889 Reinhard Zumkeller, <a href="/A097889/b097889.txt">Table of n, a(n) for n = 1..10000</a>
%F A097889 a(n) ~ n^2 log^2 n. - _Charles R Greathouse IV_, Oct 24 2012
%e A097889 1001 = 7 * 11 * 13.
%p A097889 isA097889 := proc(n)
%p A097889     local plist,p,i ;
%p A097889     plist := sort(convert(numtheory[factorset](n),list)) ;
%p A097889     if nops(plist) < 2 then
%p A097889         return false;
%p A097889     end if;
%p A097889     for i from 1 to nops(plist) do
%p A097889         p := op(i,plist) ;
%p A097889         if modp(n,p^2) = 0 then
%p A097889             return false;
%p A097889         end if;
%p A097889         if i > 1 then
%p A097889             if nextprime(op(i-1,plist)) <> p then
%p A097889                 return false;
%p A097889             end if;
%p A097889         end if;
%p A097889     end do:
%p A097889     true;
%p A097889 end proc:
%p A097889 for n from 1 to 1000 do
%p A097889     if isA097889(n) then
%p A097889         printf("%d,",n);
%p A097889     end if;
%p A097889 end do: # _R. J. Mathar_, Jan 12 2016
%t A097889 a = {}; Do[ AppendTo[a, Apply[ Times, (Prime /@ Partition[ Range[30], n, i]), 1]], {n, 2, 6}, {i, n - 1}]; Take[ Union[ Flatten[ a]], 45] (* _Robert G. Wilson v_, Sep 24 2004 *)
%o A097889 (Haskell)
%o A097889 import Data.Set (singleton, deleteFindMin, insert)
%o A097889 a097889 n = a097889_list !! (n-1)
%o A097889 a097889_list = f $ singleton (6, 2, 3) where
%o A097889    f s = y : f (insert (w, p, q') $ insert (w `div` p, a151800 p, q') s')
%o A097889          where w = y * q'; q' = a151800 q
%o A097889                ((y, p, q), s') = deleteFindMin s
%o A097889 -- _Reinhard Zumkeller_, May 12 2015, Aug 26 2011
%o A097889 (PARI) list(lim)=my(v=List(), p, t); for(e=2, log(lim+.5)\log(2), p=1; t=prod(i=1, e-1, prime(i)); forprime(q=prime(e), lim, t*=q/p; if(t>lim, next(2)); listput(v, t); p=nextprime(p+1))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Oct 24 2012
%o A097889 (Python)
%o A097889 import heapq
%o A097889 from sympy import sieve
%o A097889 sieve.extend(10**6)
%o A097889 primes = list(sieve._list)
%o A097889 def prime(n): return primes[n-1]
%o A097889 def aupton(terms, verbose=False):
%o A097889     p = prime(1)*prime(2); h = [(p, 1, 2)]; nextcount = 3; alst = []
%o A097889     while len(alst) < terms:
%o A097889         (v, s, l) = heapq.heappop(h)
%o A097889         alst.append(v)
%o A097889         if verbose: print(f"{v}, [= Prod_{{i = {s}..{l}}} prime(i)]")
%o A097889         if v >= p:
%o A097889             p *= prime(nextcount)
%o A097889             heapq.heappush(h, (p, 1, nextcount))
%o A097889             nextcount += 1
%o A097889         v //= prime(s); s += 1; l += 1; v *= prime(l)
%o A097889         heapq.heappush(h, (v, s, l))
%o A097889     return alst
%o A097889 print(aupton(45)) # _Michael S. Branicky_, Jun 15 2021
%Y A097889 Union of A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, etc.
%Y A097889 Cf. A050936.
%Y A097889 Intersection of A073485 and A002808.
%Y A097889 Cf. A151800, A215366.
%K A097889 nonn,easy
%O A097889 1,1
%A A097889 Bart la Bastide (bart(AT)xs4all.nl), Sep 21 2004
%E A097889 More terms from _Robert G. Wilson v_, Sep 24 2004
%E A097889 Data corrected for n > 41 by _Reinhard Zumkeller_, Aug 26 2011