A046325 -id:A046325 - cp's OEIS frontend

A006094 Products of 2 successive primes.

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

A046301 Product of 3 successive primes.

Original entry on oeis.org

30, 105, 385, 1001, 2431, 4199, 7429, 12673, 20677, 33263, 47027, 65231, 82861, 107113, 146969, 190747, 241133, 290177, 347261, 409457, 478661, 583573, 716539, 871933, 1009091, 1113121, 1201289, 1317919, 1564259, 1879981, 2279269, 2494633, 2837407, 3127361, 3532343

Offset: 1

Patrick De Geest, Jun 15 1998

			From _K. D. Bajpai_, Aug 27 2014: (Start)
a(2) = 105 is in the sequence because 105 = 3* 5 * 7, product of three successive primes.
a(3) = 385 is in the sequence because 385 = 5 * 7 * 11, product of three successive primes.
(End)

Vincenzo Librandi and K. D. Bajpai, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Cf. A002110, A006094, A046302, A046303, A046324, A046325, A046326, A046327.

Cf. A000040, A242187, A257891.

a046301 n = a046301_list !! (n-1)
a046301_list = zipWith3 (((*) .) . (*))
               a000040_list (tail a000040_list) (drop 2 a000040_list)
-- Reinhard Zumkeller, May 12 2015

[NthPrime(n)*NthPrime(n+1)*NthPrime(n+2): n in [1..31]]; /* Or: */ [&*[ NthPrime(n+k): k in [0..2] ]: n in [1..31] ]; // Bruno Berselli, Feb 25 2011

A046301:=n->ithprime(n)*ithprime(n+1)*ithprime(n+2): seq(A028560(n), n=1..100);  # K. D. Bajpai, Aug 27 2014

Table[Prime[n] Prime[n+1] Prime[n+2],{n,50}]   (* K. D. Bajpai, Aug 27 2014 *)
Times@@@Partition[Prime[Range[40]],3,1] (* Harvey P. Dale, Mar 25 2019 *)

a(n)=prime(n)*prime(n+1)*prime(n+2); \\ Joerg Arndt, Aug 30 2014

Sum_{n>=1} 1/a(n) = A242187. - Amiram Eldar, Nov 19 2020

Offset changed from 0 to 1 by Vincenzo Librandi, Jan 16 2012

A046303 Product of 5 successive primes.

Original entry on oeis.org

2310, 15015, 85085, 323323, 1062347, 2800733, 6678671, 14535931, 31367009, 58642669, 95041567, 162490421, 259106347, 385499687, 600662303, 907383479, 1249792339, 1673450759, 2276990377, 3024658859, 4132280413, 5717264681

Offset: 1

Patrick De Geest, Jun 15 1998

John Cerkan, Table of n, a(n) for n = 1..10000 (terms 1..520 from Vincenzo Librandi)

A subsequence of A014614.

Cf. A002110, A006094, A046301, A046302, A046324, A046325, A046326, A046327.

[&*[ NthPrime(n+k): k in [0..4] ]: n in [1..22]];  // Bruno Berselli, Feb 25 2011

lst={};Do[p0=Prime[n];p1=Prime[n+1];p2=Prime[n+2];p3=Prime[n+3];p4=Prime[n+4];a=p0*p1*p2*p3*p4;AppendTo[lst,a],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 10 2009 *)
Times@@@Partition[Prime[Range[200]],5,1] (* Harvey P. Dale, Oct 21 2011 *)

first(n)=my(P=primes(n+4)); vector(n,i,prod(j=i,i+4,P[j])) \\ Charles R Greathouse IV, Jun 27 2019

a(n) = Product_{j=n..n+4} prime(j). - Jon E. Schoenfield, Jan 07 2015

a(n) ~ (n log n)^5. - Charles R Greathouse IV, Jun 27 2019

A046302 Products of 4 successive primes.

Original entry on oeis.org

210, 1155, 5005, 17017, 46189, 96577, 215441, 392863, 765049, 1363783, 2022161, 3065857, 4391633, 6319667, 8965109, 12780049, 17120443, 21182921, 27433619, 33984931, 42600829, 56606581, 72370439, 89809099, 107972737, 121330189

Offset: 1

Patrick De Geest, Jun 15 1998

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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.

Cf. A002110, A006094, A046301, A046303, A046324, A046325, A046326, A046327.

[&*[ NthPrime(n+k): k in [0..3] ]: n in [1..26] ]; // Bruno Berselli, Feb 25 2011

lst={};Do[p0=Prime[n];p1=Prime[n+1];p2=Prime[n+2];p3=Prime[n+3];a=p0*p1*p2*p3;AppendTo[lst,a],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 10 2009 *)
Times@@@Partition[Prime[Range[50]],4,1] (* Harvey P. Dale, Sep 19 2011 *)

a(n) = Product_{j=n..n+3} prime(j). - Jon E. Schoenfield, Jan 07 2015

Offset changed from 0 to 1 by Vincenzo Librandi, Jan 16 2012

A046324 Product of 6 successive primes.

Original entry on oeis.org

30030, 255255, 1616615, 7436429, 30808063, 86822723, 247110827, 595973171, 1348781387, 2756205443, 5037203051, 9586934839, 15805487167, 25828479029, 42647023513, 66238993967, 98733594781, 138896412997, 202652143553

Offset: 1

Patrick De Geest, Jun 15 1998

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002110, A006094, A046301, A046302, A046303, A046325, A046326, A046327.

[&*[ NthPrime(n+k): k in [0..5] ]: n in [1..19] ];  // Bruno Berselli, Feb 25 2011

Times@@@Partition[Prime[Range[200]],6,1] (* Harvey P. Dale, Oct 21 2011 *)

Offset changed from 0 to 1 by Vincenzo Librandi, Jan 16 2012

A046326 Product of 8 successive primes.

Original entry on oeis.org

9699690, 111546435, 1078282205, 6685349671, 35336848261, 131710070791, 435656388001, 1204461778591, 3359814435017, 8618654420261, 18128893780549, 39181802686993, 75186702453419, 133869006807307, 245945384599471

Offset: 1

Patrick De Geest, Jun 15 1998

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002110.

Cf. product of n successive primes: A006094, A046301, A046302, A046303, A046324, A046325, A046327.

[&*[ NthPrime(n+k): k in [0..7] ]: n in [1..15]];  // Bruno Berselli, Feb 25 2011

Times@@@Partition[Prime[Range[50]],8,1] (* Harvey P. Dale, Oct 21 2011 *)

Offset changed from 0 to 1 by Vincenzo Librandi, Jan 16 2012

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

Original entry on oeis.org

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

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

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

			1001 = 7 * 11 * 13.

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

Union of A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, etc.
Cf. A050936.
Intersection of A073485 and A002808.
Cf. A151800, A215366.

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

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

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 *)

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

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

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

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

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

A127346 Primes in A127345.

Original entry on oeis.org

31, 71, 167, 311, 1151, 3119, 4871, 5711, 6791, 14831, 24071, 33911, 60167, 79031, 101159, 106367, 115631, 158231, 235751, 259751, 366791, 402551, 455471, 565919, 635711, 644951, 1124831, 1347971, 1510799, 1547927, 1743419, 1851671, 2048471

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

Primes of the form prime(k)*prime(k+1) + prime(k)*prime(k+2) + prime(k+1)*prime(k+2).

A prime number n is in the sequence if for some k it is the coefficient of x^1 of the polynomial Product_{j=0..2} (x-prime(k+j)); the roots of this polynomial are prime(k), ..., prime(k+2).

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A127345, A127347, A127351, A006094, A002110, A034962, A034965, A082246, A082251, A127340, A127341, A070934, A046301, A046302, A046303, A046324, A046325, A046326, A046327.

b = {}; a = {}; Do[If[PrimeQ[Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x + 1] Prime[x + 2]], AppendTo[a, Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x + 1] Prime[x + 2]], AppendTo[b, Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x + 1] Prime[x + 2]]], {x, 1, 100}]; Print[a] (* Artur Jasinski, Jan 11 2007 *)
s[li_] := li[[1]]*(li[[2]]+li[[3]])+li[[2]]*li[[3]]; Select[(s[#]&/@Partition[Prime[Range[100]], 3, 1]), PrimeQ] (* Zak Seidov, Jan 13 2012 *)

{m=143;k=2;for(n=1,m,a=sum(i=n,n+k-1,sum(j=i+1,n+k,prime(i)*prime(j)));if(isprime(a),print1(a,",")))} \\ Klaus Brockhaus, Jan 21 2007

{m=143;k=2;for(n=1,m,a=polcoeff(prod(j=0,k,(x-prime(n+j))),1);if(isprime(a),print1(a,",")))} \\ Klaus Brockhaus, Jan 21 2007

p=2; q=3; forprime(r=5, 1e3, if(isprime(t=p*q+p*r+q*r), print1(t", ")); p=q; q=r) \\ Charles R Greathouse IV, Jan 13 2012

a(n) = A127345(A204231(n)). - Zak Seidov, Jan 13 2012

Edited and extended by Klaus Brockhaus, Jan 21 2007

A127343 Product of 11 consecutive primes.

Original entry on oeis.org

200560490130, 3710369067405, 50708377254535, 436092044389001, 2928046583754721, 14107860812636383, 64027983688118969, 229747470880897477, 810162134158954261, 2500935283708076197

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) is the absolute value of the coefficient of x^0 of the polynomial Product_{j=0..10} (x-prime(n+j)) of degree 11; the roots of this polynomial are prime(n), ..., prime(n+10).

Cf. A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127342, A127344.

[&*[ NthPrime(n+k): k in [0..10] ]: n in [1..50] ]; // Vincenzo Librandi, Apr 03 2011

a = {}; Do[AppendTo[a, Product[Prime[x + n], {n, 0, 10}]], {x, 1, 50}]; a
Times@@@Partition[Prime[Range[50]],11,1] (* Harvey P. Dale, Oct 21 2011 *)

{m=10;k=11;for(n=0,m-1,print1(a=prod(j=1,k,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 21 2007

{m=10;k=11;for(n=1,m,print1(abs(polcoeff(prod(j=0,k-1,(x-prime(n+j))),0)),","))} \\ Klaus Brockhaus, Jan 21 2007

Edited by Klaus Brockhaus, Jan 21 2007

A096334 Triangle read by rows: T(n,k) = prime(n)#/prime(k)#, 0<=k<=n.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 30, 15, 5, 1, 210, 105, 35, 7, 1, 2310, 1155, 385, 77, 11, 1, 30030, 15015, 5005, 1001, 143, 13, 1, 510510, 255255, 85085, 17017, 2431, 221, 17, 1, 9699690, 4849845, 1616615, 323323, 46189, 4199, 323, 19, 1, 223092870, 111546435, 37182145, 7436429, 1062347, 96577, 7429, 437, 23, 1

Offset: 0

Reinhard Zumkeller, Aug 03 2004

T(n,k) is the (k+1)-th product of (n-k) successive primes (k, n-(k+1) >= 0). - Alois P. Heinz, Jan 21 2022

			Triangle begins:
    1;
    2,   1;
    6,   3,  1;
   30,  15,  5, 1;
  210, 105, 35, 7, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

T(n-1+j,n-1) give (j=1-12): A000040, A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127342, A127343, A127344.
Columns k=0-1 give: A002110, A070826.
T(2n,n) gives A107712.
Row sums give A350895.
Antidiagonal sums give A350758.
Cf. A073485 (distinct values sorted).

T:= proc(n, k) option remember;
     `if`(n=k, 1, T(n-1, k)*ithprime(n))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jan 21 2022

T[n_, k_] := Times @@ Prime[Range[k + 1, n]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 13 2021 *)

pr(n) = factorback(primes(n)); \\ A002110
row(n) = my(P=pr(n)); vector(n+1, k, P/pr(k-1)); \\ Michel Marcus, Jan 21 2022

T(n,0) = A002110(n); T(n,n) = 1;
T(n,n-1) = A000040(n) for n>0;
T(n,k) = A002110(n)/A002110(k), 0<=k<=n.
T(n,k) = Product_{j=k+1..n} prime(j). - Alois P. Heinz, Jan 21 2022

cp's OEIS Frontend

A006094 Products of 2 successive primes.

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A046301 Product of 3 successive primes.

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A046303 Product of 5 successive primes.

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A046302 Products of 4 successive primes.

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A046324 Product of 6 successive primes.

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A046326 Product of 8 successive primes.

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

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