A097889 -id:A097889 - cp's OEIS frontend

A266955 Intersection of A046346 (numbers that are divisible by the sum of their prime factors, counted with multiplicity) and A097889 (numbers that are products of at least two consecutive primes).

30, 105, 15015, 9699690, 37182145, 215656441, 955049953, 33426748355, 247357937827, 1448810778701, 3710369067405, 304250263527210, 102481630431415235, 1086305282573001491, 261682369333342226303, 37420578814667938361329, 241532826894674874877669

Offset: 1

Michel Marcus, Jan 07 2016

nonn

Alladi and Erdős ask if this sequence is infinite and give 3 terms: 2*3*5, 2*3*5*7*11*13*17*19 and 2*3*5*7*11*13*17*19*23*29*31*37*41, that is, a(1), a(4) and a(12).

This sequence contains A159578(n) for all values of n > 1. - Altug Alkan, Jan 07 2016

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..500
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294.

Cf. A046346, A097889, A159578.

sopfr(n) = {my(f=factor(n)); sum(k=1, #f~, f[k, 1]*f[k, 2]); }
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)); if (! (t % sopfr(t)), listput(v, t)); p=nextprime(p+1))); vecsort(Vec(v));} \\ adapted from A097889

a(13)-a(17) from Hiroaki Yamanouchi, Jan 12 2016

A006094 Products of 2 successive primes.

Original entry on oeis.org

6, 15, 35, 77, 143, 221, 323, 437, 667, 899, 1147, 1517, 1763, 2021, 2491, 3127, 3599, 4087, 4757, 5183, 5767, 6557, 7387, 8633, 9797, 10403, 11021, 11663, 12317, 14351, 16637, 17947, 19043, 20711, 22499, 23707, 25591, 27221, 28891, 30967, 32399, 34571, 36863

Offset: 1

N. J. A. Sloane

The Huntley reference would suggest prefixing the sequence with an initial 4 - Enoch Haga. [But that would conflict with the definition! - N. J. A. Sloane, Oct 13 2009]
Sequence appears to coincide with the sequence of numbers n such that the largest prime < sqrt(n) and the smallest prime > sqrt(n) divide n. - Benoit Cloitre, Apr 04 2002
This is true: p(n) < [ sqrt(a(n)) = sqrt(p(n)*p(n+1)) ] < p(n+1) by definition. - Jon Perry, Oct 02 2013
a(n+1) = smallest number such that gcd(a(n), a(n+1)) = prime(n+1). - Alexandre Wajnberg and Ray Chandler, Oct 14 2005
Also the area of rectangles whose side lengths are consecutive primes. E.g., the consecutive primes 7,11 produce a 7 X 11 unit rectangle which has area 77 square units. - Cino Hilliard, Jul 28 2006
a(n) = A001358(A172348(n)); A046301(n) = lcm(a(n), a(n+1)); A065091(n) = gcd(a(n), a(n+1)); A066116(n+2) = a(n+1)*a(n); A109805(n) = a(n+1) - a(n). - Reinhard Zumkeller, Mar 13 2011
See A209329 for the sum of the reciprocals. - M. F. Hasler, Jan 22 2013
A078898(a(n)) = 3. - Reinhard Zumkeller, Apr 06 2015

H. E. Huntley, The Divine Proportion, A Study in Mathematical Beauty. New York: Dover, 1970. See Chapter 13, Spira Mirabilis, especially Fig. 13-5, page 173.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
A. Bernoff and R. Pennington, Problems Drive 1984, Archimedeans Problems Drive, Eureka, 45 (1985), 22-25, 50. (Annotated scanned copy)
C. Cobeli and A. Zaharescu, A game with divisors and absolute differences of exponents, Journal of Difference Equations and Applications, Vol. 20, #11 (2014) pp. 1489-1501, DOI: 10.1080/10236198.2014.940337. Also available as arXiv:1411.1334 [math.NT], 2014.

Subset of the squarefree semiprimes, A006881.
Cf. A090076, A090090.
Cf. A166329, A152241, A030664, A219603.
Cf. A046301, A046302, A046303, A046324, A046325, A046326, A046327.
Subsequence of A256617 and A097889.
Cf. A000040, A078898.

a006094 n = a006094_list !! (n-1)
a006094_list = zipWith (*) a000040_list a065091_list
-- Reinhard Zumkeller, Mar 13 2011

a006094_list = pr a000040_list
  where pr (n:m:tail) = n*m : pr (m:tail)
        pr _ = []
-- Jean-François Antoniotti, Jan 08 2020

[NthPrime(n)*NthPrime(n+1): n in [1..41]]; // Bruno Berselli, Feb 24 2011

a:= n-> (p-> p(n)*p(n+1))(ithprime):
seq(a(n), n=1..43);  # Alois P. Heinz, Jan 02 2021

Table[ Prime[n] Prime[n + 1], {n, 40}] (* Robert G. Wilson v, Jan 22 2004 *)
Times@@@Partition[Prime[Range[60]], 2, 1] (* Harvey P. Dale, Oct 15 2011 *)

ithprime(i)*ithprime(i+1) $ i = 1..41 // Zerinvary Lajos, Feb 26 2007

g(n) = for(x=1,n,print1(prime(x)*prime(x+1)",")) \\ Cino Hilliard, Jul 28 2006

is(n)=my(p=precprime(sqrtint(n))); p>1 && n%p==0 && isprime(n/p) && nextprime(p+1)==n/p \\ Charles R Greathouse IV, Jun 04 2014

from sympy import prime, primerange
def aupton(nn):
    alst, prevp = [], 2
    for p in primerange(3, prime(nn+1)+1): alst.append(prevp*p); prevp = p
    return alst
print(aupton(43)) # Michael S. Branicky, Jun 15 2021

from sympy import prime, nextprime
def A006094(n): return (p:=prime(n))*nextprime(p) # Chai Wah Wu, Oct 18 2024

A209329 = Sum_{n>=2} 1/a(n). - M. F. Hasler, Jan 22 2013

a(n) = A000040(n) * A000040(n+1). - Alois P. Heinz, Jan 02 2021

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

Original entry on oeis.org

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

Reinhard Zumkeller, Aug 03 2002

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

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

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

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.

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

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

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 *)

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

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

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

A050936 Sum of two or more consecutive prime numbers.

Original entry on oeis.org

5, 8, 10, 12, 15, 17, 18, 23, 24, 26, 28, 30, 31, 36, 39, 41, 42, 48, 49, 52, 53, 56, 58, 59, 60, 67, 68, 71, 72, 75, 77, 78, 83, 84, 88, 90, 95, 97, 98, 100, 101, 102, 109, 112, 119, 120, 121, 124, 127, 128, 129, 131, 132, 138, 139, 143, 144, 150, 152, 155, 156, 158, 159, 160, 161, 162

Offset: 1

G. L. Honaker, Jr., Dec 31 1999

			E.g., 5 = (2 + 3) or (#2,2).
2+3 = 5, 3+5 = 8, 2+3+5 = 10, 7+5 = 12, 3+5+7 = 15, etc.

T. D. Noe, Table of n, a(n) for n = 1..10000
Patrick De Geest, WONplate 122
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.
Eric Weisstein's World of Mathematics, Prime Sums

Subsequence of A034707.
Cf. A067372 up to A067381, A054996, A000040.
A084143(a(n)) > 0, complement of A087072.
Cf. A054845, A097889.

import Data.Set (empty, findMin, deleteMin, insert)
import qualified Data.Set as Set (null)
a050936 n = a050936_list !! (n-1)
a050936_list = f empty [2] 2 $ tail a000040_list where
   f s bs c (p:ps)
     | Set.null s || head bs <= m = f (foldl (flip insert) s bs') bs' p ps
     | otherwise                  = m : f (deleteMin s) bs c (p:ps)
     where m = findMin s
           bs' = map (+ p) (c : bs)
-- Reinhard Zumkeller, Aug 26 2011

# uses code of A084143
isA050936 := proc(n::integer)
    if A084143(n) >= 1 then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 300 do
    if isA050936(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Aug 19 2020

lst={};Do[p=Prime[n];Do[p=p+Prime[k];AppendTo[lst, p], {k, n+1, 2*10^2}], {n, 2*10^2}];Take[Union[lst], 10^2] (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
f[n_] := Block[{len = PrimePi@ n}, p = Prime@ Range@ len; Count[ Flatten[ Table[ p[[i ;; j]], {i, len}, {j, i+1, len}],1], q_ /; Total@ q == n]]; Select[ Range@ 150, f@ # > 0 &] (* Or quicker for a larger range *)
lmt = 150; p = Prime@ Range@ PrimePi@ lmt; t = Table[0, {lmt}]; Do[s = 0; j = i+1; While[s = s + p[[j]]; s <= lmt, t[[s]]++; j++], {i, Length@ p}]; Select[ Range@ lmt, t[[#]] > 0 &] (* Robert G. Wilson v *)
Module[{nn=70,prs},prs=Prime[Range[nn]];Take[Union[Flatten[Table[Total/@ Partition[prs,i,1],{i,2,nn}]]],nn]] (* Harvey P. Dale, Nov 13 2013 *)

is(n)=my(v,m=1,t); while(1,v=vector(m++); v[m\2]=precprime(n\m); for(i=m\2+1,m, v[i]=nextprime(v[i-1]+1)); forstep(i=m\2-1,1,-1, v[i]=precprime(v[i+1]-1)); if(v[1]==0, return(0)); t=vecsum(v); if(t==n,return(1)); if(t>n, while(t>n,t-=v[m]; v=concat(precprime(v[1]-1), v[1..m-1]); t+=v[1]), while(tCharles R Greathouse IV, May 05 2016

list(lim)=my(v=List(),s,n=1,p); while(1, p=2; s=vecsum(primes(n++)); if(s>lim,return(Set(v))); listput(v,s); forprime(q=prime(n+1),, s+=q-p; if(s>lim,break); listput(v,s); p=nextprime(p+1))); \\ Charles R Greathouse IV, Nov 24 2021

More terms from David W. Wilson, Jan 13 2000

A066312 Numbers, other than prime powers, whose distinct prime factors are consecutive primes.

Original entry on oeis.org

6, 12, 15, 18, 24, 30, 35, 36, 45, 48, 54, 60, 72, 75, 77, 90, 96, 105, 108, 120, 135, 143, 144, 150, 162, 175, 180, 192, 210, 216, 221, 225, 240, 245, 270, 288, 300, 315, 323, 324, 360, 375, 384, 385, 405, 420, 432, 437, 450, 480, 486

Offset: 1

Leroy Quet, Jan 01 2002

nonn

Numbers whose squarefree kernel (A007947) is the product of 2 or more consecutive primes. - Peter Munn, May 27 2023

			75 is included because 75 = 3 * 5^2 and 3 and 5 are consecutive primes.

Harry J. Smith, Table of n, a(n) for n=1..1000

Cf. A007947, A097889.

Select[Range[2, 500], And[! PrimePowerQ@ #, Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}] &] (* Michael De Vlieger, Sep 24 2017 *)

{ n=0; for (m=2, 10^9, f=factor(m); b=1; if (matsize(f)[1] == 1, next); for (i=2, matsize(f)[1], if (primepi(f[i, 1]) - primepi(f[i - 1, 1]) > 1, b=0; break)); if (b, write("b066312.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 10 2010

{a(n) : n >= 1} = {k >= 1 : A007947(k) is in A097889}. - Peter Munn, May 29 2023

OFFSET changed from 0,1 to 1,1 by Harry J. Smith, Feb 10 2010

Name edited by Peter Munn, May 29 2023

A045619 Numbers that are the products of 2 or more consecutive integers.

Original entry on oeis.org

0, 2, 6, 12, 20, 24, 30, 42, 56, 60, 72, 90, 110, 120, 132, 156, 182, 210, 240, 272, 306, 336, 342, 360, 380, 420, 462, 504, 506, 552, 600, 650, 702, 720, 756, 812, 840, 870, 930, 990, 992, 1056, 1122, 1190, 1260, 1320, 1332, 1406, 1482, 1560, 1640, 1680

Offset: 1

Erich Friedman

Erdős and Selfridge proved that, apart from the first term, these are never perfect powers (A001597). - T. D. Noe, Oct 13 2002

Numbers of the form x!/y! with y+1 < x. - Reinhard Zumkeller, Feb 20 2008

			30 is in the sequence as 30 = 5*6 = 5*(5+1). - _David A. Corneth_, Oct 19 2021

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T.D. Noe)
P. Erdős and J. L. Selfridge, The product of consecutive integers is never a power, Illinois Jour. Math. 19 (1975), 292-301.

Union of A002378, A007531, A052762, A052787, A053625, etc. - R. J. Mathar, Oct 19 2021

Cf. A001597, A000142, A137895, A093449, A064224, A084720, A137899, A137900, A120436, A097889.

maxNum = 1700; lst = {}; For[i = 1, i <= Sqrt[maxNum], i++, j = i + 1; prod = i*j; While[prod < maxNum, AppendTo[lst, prod]; j++; prod *= j]]; lst = Union[lst]

list(lim)=my(v=List([0]),P,k=1,t); while(1, k++; P=binomial('n+k-1,k)*k!; if(subst(P,'n,1)>lim, break); for(n=1,lim, t=eval(P); if(t>lim, next(2)); listput(v,t))); Set(v) \\ Charles R Greathouse IV, Nov 16 2021

import heapq
from sympy import sieve
def aupton(terms, verbose=False):
    p = 6; h = [(p, 2, 3)]; nextcount = 4; aset = {0, 2}
    while len(aset) < terms:
        (v, s, l) = heapq.heappop(h)
        aset.add(v)
        if verbose: print(f"{v}, [= Prod_{{i = {s}..{l}}} i]")
        if v >= p:
            p *= nextcount
            heapq.heappush(h, (p, 2, nextcount))
            nextcount += 1
        v //= s; s += 1; l += 1; v *= l
        heapq.heappush(h, (v, s, l))
    return sorted(aset)
print(aupton(52)) # Michael S. Branicky, Oct 19 2021

a(n) = A000142(A137911(n))/A000142(A137912(n)-1) for n>1. - Reinhard Zumkeller, Feb 27 2008
Since the oblong numbers (A002378) have relative density of 100%, we have a(n) ~ (n-1) n ~ n^2. - Daniel Forgues, Mar 26 2012
a(n) = n^2 - 2*n^(5/3) + O(n^(4/3)). - Charles R Greathouse IV, Aug 27 2013

More terms from Larry Reeves (larryr(AT)acm.org), Jul 20 2000
More terms from Reinhard Zumkeller, Feb 27 2008
Incorrect program removed by David A. Corneth, Oct 19 2021

A073490 Number of prime gaps in factorization of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0

Offset: 1

Reinhard Zumkeller, Aug 03 2002

A137723(n) is the smallest number of the first occurring set of exactly n consecutive numbers with at least one prime gap in their factorization: a(A137723(n)+k)>0 for 0<=kA137723(n)-1)=a(A137723(n)+n)=0. - Reinhard Zumkeller, Feb 09 2008

			84 = 2*2*3*7 with one gap between 3 and 7, therefore a(84) = 1;
110 = 2*5*11 with two gaps: between 2 and 5 and between 5 and 11, therefore a(110) = 2.

R. Zumkeller, Table of n, a(n) for n = 1..50000

Cf. A073491, A073492, A073493, A073494, A073495.

Cf. A137721, A137722, A027748, A049084.

a073490 1 = 0
a073490 n = length $ filter (> 1) $ zipWith (-) (tail ips) ips
   where ips = map a049084 $ a027748_row n
-- Reinhard Zumkeller, Jul 04 2012

A073490 := proc(n)
    local a,plist ;
    plist := sort(convert(numtheory[factorset](n),list)) ;
    a := 0 ;
    for i from 2 to nops(plist) do
        if op(i,plist) <> nextprime(op(i-1,plist)) then
            a := a+1 ;
        end if;
    end do:
    a;
end proc:
seq(A073490(n),n=1..110) ; # R. J. Mathar, Oct 27 2019

gaps[n_Integer/;n>0]:=If[n===1, 0, Complement[Prime[PrimePi[Rest[ # ]]-1], # ]&[First/@FactorInteger[n]]]; Table[Length[gaps[n]], {n, 1, 120}] (* Wouter Meeussen, Oct 30 2004 *)
pa[n_, k_] := If[k == NextPrime[n], 0, 1]; Table[Total[pa @@@ Partition[First /@ FactorInteger[n], 2, 1]], {n, 120}] (* Jayanta Basu, Jul 01 2013 *)

from sympy import primefactors, nextprime
def a(n):
    pf = primefactors(n)
    return sum(p2 != nextprime(p1) for p1, p2 in zip(pf[:-1], pf[1:]))
print([a(n) for n in range(1, 121)]) # Michael S. Branicky, Oct 14 2021

a(n) = A073484(A007947(n)).
a(A000040(n))=0; a(A000961(n))=0; a(A006094(n))=0; a(A002110(n))=0; a(A073485(n))=0.
a(A073486(n))>0; a(A073487(n)) = 1; a(A073488(n))=2; a(A073489(n))=3.
a(n)=0 iff A073483(n) = 1.
a(A097889(n)) = 0. - Reinhard Zumkeller, Nov 20 2004
0 <= a(m*n) <= a(m) + a(n) + 1. A137794(n) = 0^a(n). - Reinhard Zumkeller, Feb 11 2008

More terms from Franklin T. Adams-Watters, May 19 2006

A378746 a(n) = Product{i=n..primepi(2*prime(n))} prime(i).

Original entry on oeis.org

6, 15, 35, 1001, 46189, 96577, 6678671, 14535931, 1348781387, 146078888479, 18128893780549, 203079283326684719, 433601713048867373, 877779077635511999, 1816798556036292277, 39006703653387621491281, 969956148531489825059765363, 16439934720872708899318057, 4483790064773102589474664169, 1274400211992152128527851190601

Offset: 1

Antti Karttunen, Dec 08 2024

nonn

Each a(n) is a proper divisor of A007741(n).
Each a(n) or one of its proper divisors is present in A337372.
a(n) is the least product of consecutive primes starting from A000040(n) such that A003961(a(n)) > 2*a(n). Note that 6, 15, 35, 46189 and 96577 are all present in A337372, i.e., are primitively primeshift-abundant numbers. For 1001 = 7*11*31, it is however its proper divisor 91 = 7*31 which gets that honor.

Antti Karttunen, Table of n, a(n) for n = 1..344

Cf. A003961, A007741, A063124 [= A001222(a(n))], A337372.
Subsequence of A097889.
Cf. also A378745.

A378746(n) = prod(i=n,primepi(2*prime(n)),prime(i));

A257836 Numbers which are the product of at least two consecutive odd numbers > 1.

Original entry on oeis.org

15, 35, 63, 99, 105, 143, 195, 255, 315, 323, 399, 483, 575, 675, 693, 783, 899, 945, 1023, 1155, 1287, 1295, 1443, 1599, 1763, 1935, 2115, 2145, 2303, 2499, 2703, 2915, 3135, 3315, 3363, 3465, 3599, 3843, 4095, 4355, 4623, 4845, 4899, 5183, 5475, 5775, 6083

Offset: 1

Reinhard Zumkeller, May 12 2015

nonn

			.     |       |   ----- Factorizations into ... --------------
.   n |  a(n) |   prime powers     |   consecutive odd numbers
. ----+-------+--------------------+--------------------------
.   1 |    15 |   3 * 5            |   3 * 5
.   2 |    35 |   5 * 7            |   5 * 7
.   3 |    63 |   3^2 * 7          |   7 * 9
.   4 |    99 |   3^2 * 11         |   9 * 11
.   5 |   105 |   3 * 5 * 7        |   3 * 5 * 7
.   6 |   143 |   11 * 13          |   11 * 13
.   7 |   195 |   3 * 5 * 13       |   13 * 15
.   8 |   255 |   3 * 5 * 17       |   15 * 17
.   9 |   315 |   3^2 * 5 * 7      |   5 * 7 * 9
.  10 |   323 |   17 * 19          |   17 * 19
.  11 |   399 |   3 * 7 * 19       |   19 * 21
.  12 |   483 |   3 * 7 * 23       |   21 * 23
.  13 |   575 |   5^2 * 23         |   23 * 25
.  14 |   675 |   3^3 * 5^2        |   25 * 27
.  15 |   693 |   3^2 * 7 * 11     |   7 * 9 * 11
.  16 |   783 |   3^3 * 29         |   27 * 29
.  17 |   899 |   29 * 31          |   29 * 31
.  18 |   945 |   3^3 * 5 * 7      |   3 * 5 * 7 * 9
.  19 |  1023 |   3 * 11 * 31      |   31 * 33
.  20 |  1155 |   3 * 5 * 7 * 11   |   33 * 35
.  21 |  1287 |   3^2 * 11 * 13    |   9 * 11 * 13
.  22 |  1295 |   5 * 7 * 37       |   35 * 37
.  23 |  1443 |   3 * 13 * 37      |   37 * 39
.  24 |  1599 |   3 * 13 * 41      |   39 * 41
.  25 |  1763 |   41 * 43          |   41 * 43
.  26 |  1935 |   3^2 * 5 * 43     |   43 * 45
.  27 |  2115 |   3^2 * 5 * 47     |   45 * 47
.  28 |  2145 |   3 * 5 * 11 * 13  |   11 * 13 * 15
.  29 |  2303 |   7^2 * 47         |   47 * 49
.  30 |  2499 |   3 * 7^2 * 17     |   49 * 51  .

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

Cf. A005408, A097889.

import Data.Set (singleton, deleteFindMin, insert)
a257836 n = a257836_list !! (n-1)
a257836_list = f $ singleton (15, 3, 5) where
   f s = y : f (insert (w, u, v') $ insert (w `div` u, u + 2, v') s')
         where w = y * v'; v' = v + 2
               ((y, u, v), s') = deleteFindMin s

A257891 Numbers that are products of at least three consecutive primes.

Original entry on oeis.org

30, 105, 210, 385, 1001, 1155, 2310, 2431, 4199, 5005, 7429, 12673, 15015, 17017, 20677, 30030, 33263, 46189, 47027, 65231, 82861, 85085, 96577, 107113, 146969, 190747, 215441, 241133, 255255, 290177, 323323, 347261, 392863, 409457, 478661, 510510, 583573

Offset: 1

Reinhard Zumkeller, May 12 2015

nonn

			a(5) = 1001 = 7 * 11 * 13;
a(6) = 1155 = 3 * 5 * 7 * 11;
a(7) = 2310 = 2 * 3 * 5 * 7 * 11;
a(8) = 2431 = 11 * 13 * 17.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A151800, A097889, A000977, A046301 (subsequence).

import Data.Set (singleton, deleteFindMin, insert)
a257891 n = a257891_list !! (n-1)
a257891_list = f $ singleton (30, 2, 5) 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

Select[Module[{nn=1000},Flatten[Table[Times@@@Partition[Prime[Range[nn]],d,1],{d,3,7}]]]//Union,#<10^7&] (* Harvey P. Dale, Aug 04 2024 *)

cp's OEIS Frontend

A266955 Intersection of A046346 (numbers that are divisible by the sum of their prime factors, counted with multiplicity) and A097889 (numbers that are products of at least two consecutive primes).

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A006094 Products of 2 successive primes.

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

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A050936 Sum of two or more consecutive prime numbers.

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A066312 Numbers, other than prime powers, whose distinct prime factors are consecutive primes.

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A045619 Numbers that are the products of 2 or more consecutive integers.

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