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A097889 Numbers that are products of (at least two) consecutive primes. 10
6, 15, 30, 35, 77, 105, 143, 210, 221, 323, 385, 437, 667, 899, 1001, 1147, 1155, 1517, 1763, 2021, 2310, 2431, 2491, 3127, 3599, 4087, 4199, 4757, 5005, 5183, 5767, 6557, 7387, 7429, 8633, 9797, 10403, 11021, 11663, 12317, 12673, 14351, 15015, 16637, 17017 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Subsequence of A073485; A073490(a(n)) = 0. - Reinhard Zumkeller, Nov 20 2004

A proper subset of A073485. - Robert G. Wilson v, Jun 11 2010

a(A192280(n)) * (1 - A010051(a(n)) = 1.

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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

a(n) ~ n^2 log^2 n. - Charles R Greathouse IV, Oct 24 2012

EXAMPLE

1001 = 7 * 11 * 13.

MAPLE

isA097889 := proc(n)

    local plist, p, i ;

    plist := sort(convert(numtheory[factorset](n), list)) ;

    if nops(plist) < 2 then

        return false;

    end if;

    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 isA097889(n) then

        printf("%d, ", n);

    end if;

end do: # R. J. Mathar, Jan 12 2016

MATHEMATICA

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 *)

PROG

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a097889 n = a097889_list !! (n-1)

a097889_list = f $ singleton (6, 2, 3) where

   f s = y : f (insert (w, p, q') $ insert (w `div` p, a151800 p, q') s')

         where w = y * q'; q' = a151800 q

               ((y, p, q), s') = deleteFindMin s

-- Reinhard Zumkeller, May 12 2015, Aug 26 2011

(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

CROSSREFS

Union of A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, etc.

Cf. A050936.

Intersection of A073485 and A002808.

Cf. A151800, A215366.

Sequence in context: A024972 A048749 A336905 * A256874 A250121 A024802

Adjacent sequences:  A097886 A097887 A097888 * A097890 A097891 A097892

KEYWORD

nonn,easy

AUTHOR

Bart la Bastide (bart(AT)xs4all.nl), Sep 21 2004

EXTENSIONS

More terms from Robert G. Wilson v, Sep 24 2004

Data corrected for n > 41 by Reinhard Zumkeller, Aug 26 2011

STATUS

approved

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Last modified September 20 20:02 EDT 2020. Contains 337265 sequences. (Running on oeis4.)