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A073485 Product of any number of consecutive primes; squarefree numbers with no gaps in their prime factorization.

%I A073485 #61 Aug 06 2022 19:07:11
%S A073485 1,2,3,5,6,7,11,13,15,17,19,23,29,30,31,35,37,41,43,47,53,59,61,67,71,
%T A073485 73,77,79,83,89,97,101,103,105,107,109,113,127,131,137,139,143,149,
%U A073485 151,157,163,167,173,179,181,191,193,197,199,210,211,221,223,227,229,233
%N A073485 Product of any number of consecutive primes; squarefree numbers with no gaps in their prime factorization.
%C A073485 A073484(a(n)) = 0 and A073483(a(n)) = 1;
%C A073485 See A097889 for composite terms. - _Reinhard Zumkeller_, Mar 30 2010
%C A073485 A169829 is a subsequence. - _Reinhard Zumkeller_, May 31 2010
%C A073485 a(A192280(n)) = 1: complement of A193166.
%C A073485 Also fixed points of A053590: a(n) = A053590(a(n)). - _Reinhard Zumkeller_, May 28 2012
%C A073485 The Heinz numbers of the partitions into distinct consecutive integers. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product_{j=1..r} prime(p_j) (concept used by _Alois P. Heinz_ in A215366 as an "encoding" of a partition). Example: (i) 15 (= 3*5) is in the sequence because it is the Heinz number of the partition [2,3]; (ii) 10 (= 2*5) is not in the sequence because it is the Heinz number of the partition [1,3]. - _Emeric Deutsch_, Oct 02 2015
%C A073485 Except for the term 1, each term can uniquely represented as A002110(k)/A002110(m) for k > m >= 0; 1 = A002110(k)/A002110(k) for all k. - _Michel Marcus_ and _Jianing Song_, Jun 19 2019
%H A073485 Alois P. Heinz, <a href="/A073485/b073485.txt">Table of n, a(n) for n = 1..20000</a> (first 1000 terms from T. D. Noe)
%F A073485 a(n) ~ n log n. - _Charles R Greathouse IV_, Oct 24 2012
%e A073485 105 is a term, as 105 = 3*5*7 with consecutive prime factors.
%p A073485 isA073485 := proc(n)
%p A073485     local plist,p,i ;
%p A073485     plist := sort(convert(numtheory[factorset](n),list)) ;
%p A073485     for i from 1 to nops(plist) do
%p A073485         p := op(i,plist) ;
%p A073485         if modp(n,p^2) = 0 then
%p A073485             return false;
%p A073485         end if;
%p A073485         if i > 1 then
%p A073485             if nextprime(op(i-1,plist)) <> p then
%p A073485                 return false;
%p A073485             end if;
%p A073485         end if;
%p A073485     end do:
%p A073485     true;
%p A073485 end proc:
%p A073485 for n from 1 to 1000 do
%p A073485     if isA073485(n) then
%p A073485         printf("%d,",n);
%p A073485     end if;
%p A073485 end do: # _R. J. Mathar_, Jan 12 2016
%p A073485 # second Maple program:
%p A073485 q:= proc(n) uses numtheory; n=1 or issqrfree(n) and (s->
%p A073485       nops(s)=1+pi(max(s))-pi(min(s)))(factorset(n))
%p A073485     end:
%p A073485 select(q, [$1..288])[];  # _Alois P. Heinz_, Jan 27 2022
%t A073485 f[n_] := FoldList[ Times, 1, Prime[ Range[n, n + 3]]]; lst = {}; k = 1; While[k < 55, AppendTo[lst, f@k]; k++ ]; Take[ Union@ Flatten@ lst, 65] (* _Robert G. Wilson v_, Jun 11 2010 *)
%o A073485 (Haskell)
%o A073485 a073485 n = a073485_list !! (n-1)
%o A073485 a073485_list = filter ((== 1) . a192280) [1..]
%o A073485 -- _Reinhard Zumkeller_, May 28 2012, Aug 26 2011
%o A073485 (PARI) list(lim)=my(v=List(primes(primepi(lim))),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
%Y A073485 Complement: A193166.
%Y A073485 Intersection of A005117 and A073491.
%Y A073485 Subsequence of A277417.
%Y A073485 Cf. A053590, A073483, A073484, A073486, A192280.
%Y A073485 Cf. A000040, A006094, A002110, A097889, A169829 (subsequences).
%Y A073485 Cf. A096334.
%K A073485 nonn,nice
%O A073485 1,2
%A A073485 _Reinhard Zumkeller_, Aug 03 2002
%E A073485 Alternative description added to the name by _Antti Karttunen_, Oct 29 2016