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

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, 73, 77, 79, 83, 89, 97, 101, 103, 105, 107, 109, 113, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 210, 211, 221, 223, 227, 229, 233

Offset: 1

Author

Reinhard Zumkeller, Aug 03 2002

Keywords

• nonn
• nice

Comments

A073484(a(n)) = 0 and A073483(a(n)) = 1;

See A097889 for composite terms. - Reinhard Zumkeller, Mar 30 2010

A169829 is a subsequence. - Reinhard Zumkeller, May 31 2010

a(A192280(n)) = 1: complement of A193166.

Also fixed points of A053590: a(n) = A053590(a(n)). - Reinhard Zumkeller, May 28 2012

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

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

Examples

			105 is a term, as 105 = 3*5*7 with consecutive prime factors.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)

Crossrefs

Complement: A193166.

Intersection of A005117 and A073491.

Subsequence of A277417.

Cf. A053590, A073483, A073484, A073486, A192280.

Cf. A000040, A006094, A002110, A097889, A169829 (subsequences).

Cf. A096334.

Programs

Haskell

a073485 n = a073485_list !! (n-1)
a073485_list = filter ((== 1) . a192280) [1..]
-- Reinhard Zumkeller, May 28 2012, Aug 26 2011

Maple

isA073485 := proc(n)
    local plist,p,i ;
    plist := sort(convert(numtheory[factorset](n),list)) ;
    for i from 1 to nops(plist) do
        p := op(i,plist) ;
        if modp(n,p^2) = 0 then
            return false;
        end if;
        if i > 1 then
            if nextprime(op(i-1,plist)) <> p then
                return false;
            end if;
        end if;
    end do:
    true;
end proc:
for n from 1 to 1000 do
    if isA073485(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Jan 12 2016
# second Maple program:
q:= proc(n) uses numtheory; n=1 or issqrfree(n) and (s->
      nops(s)=1+pi(max(s))-pi(min(s)))(factorset(n))
    end:
select(q, [$1..288])[];  # Alois P. Heinz, Jan 27 2022

Mathematica

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 *)

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

Formula

a(n) ~ n log n. - Charles R Greathouse IV, Oct 24 2012

Extensions

Alternative description added to the name by Antti Karttunen, Oct 29 2016