A046302 -id:A046302 - 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

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

A046327 Numbers that are the product of 9 successive primes.

Original entry on oeis.org

223092870, 3234846615, 33426748355, 247357937827, 1448810778701, 5663533044013, 20475850236047, 63836474265323, 198229051666003, 525737919635921, 1214635883296783, 2781907990776503, 5488629279099587

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, A046326.

[&*[ NthPrime(n+k): k in [0..8] ]: n in [1..13]];  // Bruno Berselli, Feb 25 2011

Table[Product[Prime[x+n], {n, 0, 8}], {x, 100}] (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)
Times@@@Partition[Prime[Range[25]],9,1]  (* Harvey P. Dale, Mar 05 2011 and Zak Seidov, Feb 09 2012 *)

PARI
```
a(n)=prod(k=0,8,prime(n+k))
```

a(n) = Product_{j=n..n+8} prime(j). - Jon E. Schoenfield, Jan 07 2015

More terms from Vladimir Joseph Stephan Orlovsky, Aug 26 2008

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

A046325 Product of 7 successive primes.

Original entry on oeis.org

510510, 4849845, 37182145, 215656441, 955049953, 3212440751, 10131543907, 25626846353, 63392725189, 146078888479, 297194980009, 584803025179, 1058967640189, 1833822011059, 3113232716449, 5232880523393, 8194888366823

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, A046326, A046327.

[&*[ NthPrime(n+k): k in [0..6] ]: n in [1..17]];  // Bruno Berselli, Feb 25 2011

Times@@@Partition[Prime[Range[50]],7,1] (* 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

A255483 Infinite square array read by antidiagonals downwards: T(0,m) = prime(m), m >= 1; for n >= 1, T(n,m) = T(n-1,m)*T(n-1,m+1)/gcd(T(n-1,m), T(n-1,m+1))^2, m >= 1.

Original entry on oeis.org

2, 3, 6, 5, 15, 10, 7, 35, 21, 210, 11, 77, 55, 1155, 22, 13, 143, 91, 5005, 39, 858, 17, 221, 187, 17017, 85, 3315, 1870, 19, 323, 247, 46189, 133, 11305, 5187, 9699690, 23, 437, 391, 96577, 253, 33649, 21505, 111546435, 46

Offset: 0

N. J. A. Sloane, Feb 28 2015

The first column of the array is given by A123098; subsequent columns are obtained by applying the function A003961, i.e., replacing each prime factor by the next larger prime. - M. F. Hasler, Sep 17 2016
Interpretation with respect to A329329 from Peter Munn, Feb 08 2020: (Start)
With respect to the ring defined by A329329 and A059897, the first row gives powers of 3, the first column gives powers of 6, both in order of increasing exponent, and the body of the table gives their products. A329049 is the equivalent table in which the first column gives powers of 4.
A099884 is the equivalent table for the ring defined by A048720 and A003987. That ring is an image of the polynomial ring GF(2)[x] using a standard representation of the polynomials as integers. A329329 describes a comparable mapping to integers from the related polynomial ring GF(2)[x,y].
Using these mappings, the tables here and in A099884 are matching images: the first row represents powers of x, the first column represents powers of (x+1) and the body of the table gives their products.
Hugo van der Sanden's formula (see formula section) indicates that A019565 provides a mapping from A099884. In the wider terms described above, A019565 is an injective homomorphism between images of the 2 polynomial rings, and maps the image of each GF(2)[x] polynomial to the image of the equivalent GF(2)[x,y] polynomial.
(End)

			The top left corner of the array, row index 0..5, column index 1..10:
    2,    3,     5,     7,    11,     13,     17,     19,      23,      29
    6,   15,    35,    77,   143,    221,    323,    437,     667,     899
   10,   21,    55,    91,   187,    247,    391,    551,     713,    1073
  210, 1155,  5005, 17017, 46189,  96577, 215441, 392863,  765049, 1363783
   22,   39,    85,   133,   253,    377,    527,    703,     943,    1247
  858, 3315, 11305, 33649, 95381, 198679, 370481, 662929, 1175921, 1816879

Alois P. Heinz, Antidiagonals n = 0..125, flattened
C. Cobeli, A. Zaharescu, A game with divisors and absolute differences of exponents, Journal of Difference Equations and Applications, Vol. 20, #11, 2014.
C. Cobeli, A. Zaharescu, A game with divisors and absolute differences of exponents, arXiv:1411.1334 [math.NT], 2014.
Discussion of SeqFan-mailing list
Index entries for sequences related to Gilbreath conjecture and transform

First two columns = A123098, A276804.
Rows = A000040, A006094, A090076, A046302, ...
A kind of generalization of A036262.
Cf. A003961, A007913, A019565, A048675, A066117, A099884.
Transpose: A276578, terms sorted into ascending order: A276579.
A003987, A048720, A059897, A329049 relate to the A329329 polynomial ring interpretation.

T:= proc(n, m) option remember; `if`(n=0, ithprime(m),
      T(n-1, m)*T(n-1, m+1)/igcd(T(n-1, m), T(n-1, m+1))^2)
    end:
seq(seq(T(n, 1+d-n), n=0..d), d=0..10);  # Alois P. Heinz, Feb 28 2015

T[n_, m_] := T[n, m] = If[n == 0, Prime[m], T[n-1, m]*T[n-1, m+1]/GCD[T[n-1, m], T[n-1, m+1]]^2]; Table[Table[T[n, 1+d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

T=matrix(N=15,N);for(j=1,N,T[1,j]=prime(j));(f(x,y)=x*y/gcd(x,y)^2);for(k=1,N-1,for(j=1,N-k,T[k+1,j]=f(T[k,j],T[k,j+1])));A255483=concat(vector(N,i,vector(i,j,T[j,1+i-j]))) \\ M. F. Hasler, Sep 17 2016

A255483(n,k)=prod(j=0,n,if(bitand(n-j,j),1,prime(j+k))) \\ M. F. Hasler, Sep 18 2016

(define (A255483 n) (A255483bi (A002262 n) (+ 1 (A025581 n))))
;; Then use either an almost standalone version (requiring only A000040):
(define (A255483bi row col) (if (zero? row) (A000040 col) (let ((a (A255483bi (- row 1) col)) (b (A255483bi (- row 1) (+ col 1)))) (/ (lcm a b) (gcd a b)))))
;; Or one based on M. F. Hasler's new recurrence:
(define (A255483bi row col) (if (= 1 col) (A123098 row) (A003961 (A255483bi row (- col 1)))))
;; Antti Karttunen, Sep 18 2016

T(n,1) = A123098(n), T(n,m+1) = A003961(T(n,m)), for all n >= 0, m >= 1. - M. F. Hasler, Sep 17 2016
T(n,m) = Prod_{k=0..n} prime(k+m)^(!(n-k & k)) where !x is 1 if x=0 and 0 else, and & is binary AND. - M. F. Hasler, Sep 18 2016
From Antti Karttunen, Sep 18 2016: (Start)
For n >= 1, m >= 1, T(n,m) = lcm(T(n-1,m),T(n-1,m+1)) / gcd(T(n-1,m),T(n-1,m+1)).
T(n,k) = A007913(A066117(n+1,k)).
T(n,k) = A019565(A099884(n,k-1)) [After Hugo van der Sanden's observations on SeqFan-list].
(End)
From Peter Munn, Jan 08 2020: (Start)
T(0,1) = 2, and for n >= 0, k >= 1, T(n+1,k) = A329329(T(n,k), 6), T(n,k+1) = A329329(T(n,k), 3).
T(n,k) = A329329(T(n,1), T(0,k)).
(End)

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

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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A046324 Product of 6 successive primes.

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A046326 Product of 8 successive primes.

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A046327 Numbers that are the product of 9 successive primes.

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A046325 Product of 7 successive primes.