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

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

Offset: 1

Author

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

Keywords

• nonn
• easy

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

A192280(a(n)) * (1 - A010051(a(n))) = 1. - Reinhard Zumkeller, Aug 26 2011 [corrected by Jason Yuen, Aug 29 2024]

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

Examples

			1001 = 7 * 11 * 13.
		

Links

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

Crossrefs

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

Cf. A050936.

Intersection of A073485 and A002808.

Cf. A151800, A215366.

Programs

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

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 *)

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

Python

import heapq
from sympy import sieve
sieve.extend(10**6)
primes = list(sieve._list)
def prime(n): return primes[n-1]
def aupton(terms, verbose=False):
    p = prime(1)*prime(2); h = [(p, 1, 2)]; nextcount = 3; alst = []
    while len(alst) < terms:
        (v, s, l) = heapq.heappop(h)
        alst.append(v)
        if verbose: print(f"{v}, [= Prod_{{i = {s}..{l}}} prime(i)]")
        if v >= p:
            p *= prime(nextcount)
            heapq.heappush(h, (p, 1, nextcount))
            nextcount += 1
        v //= prime(s); s += 1; l += 1; v *= prime(l)
        heapq.heappush(h, (v, s, l))
    return alst
print(aupton(45)) # Michael S. Branicky, Jun 15 2021

Formula

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

Extensions

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

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