A046301 -id:A046301 - 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

A348717 a(n) is the least k such that A003961^i(k) = n for some i >= 0 (where A003961^i denotes the i-th iterate of A003961).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 10, 2, 12, 2, 14, 6, 16, 2, 18, 2, 20, 10, 22, 2, 24, 4, 26, 8, 28, 2, 30, 2, 32, 14, 34, 6, 36, 2, 38, 22, 40, 2, 42, 2, 44, 12, 46, 2, 48, 4, 50, 26, 52, 2, 54, 10, 56, 34, 58, 2, 60, 2, 62, 20, 64, 14, 66, 2, 68, 38, 70, 2, 72, 2

Offset: 1

Rémy Sigrist, Oct 31 2021

nonn

All terms except a(1) = 1 are even.
To compute a(n) for n > 1:
- if n = Product_{j = 1..o} prime(p_j)^e_j (where prime(i) denotes the i-th prime number, p_1 < ... < p_o and e_1 > 0)
- then a(n) = Product_{j = 1..o} prime(p_j + 1 - p_1)^e_j.
This sequence has similarities with A304776: here we shift down prime indexes, there prime exponents.
Smallest number generated by uniformly decrementing the indices of the prime factors of n. Thus, for n > 1, the smallest m > 1 such that the first differences of the indices of the ordered prime factors (including repetitions) are the same for m and n. As a function, a(.) preserves properties such as prime signature. - Peter Munn, May 12 2022

Positions of particular values (see formula section): A000040, A001248, A006094, A030078, A030514, A046301, A050997, A090076, A090090, A166329, A251720.
Also see formula section for the relationship to: A000265, A003961, A004277, A005940, A020639, A046523, A055396, A071364, A122111, A156552, A243055, A243074, A297845, A322993.
Sequences with comparable definitions: A304776, A316437.
Cf. A246277 (terms halved), A305897 (restricted growth sequence transform), A354185 (Möbius transform), A354186 (Dirichlet inverse), A354187 (sum with it).

a[1] = 1; a[n_] := Module[{f = FactorInteger[n], d}, d = PrimePi[f[[1, 1]]] - 1; Times @@ ((Prime[PrimePi[#[[1]]] - d]^#[[2]]) & /@ f)]; Array[a, 100] (* Amiram Eldar, Oct 31 2021 *)

a(n) = { my (f=factor(n)); if (#f~>0, my (pi1=primepi(f[1,1])); for (k=1, #f~, f[k,1] = prime(primepi(f[k,1])-pi1+1))); factorback(f) }

a(n) = n iff n belongs to A004277.
A003961^(A055396(n)-1)(a(n)) = n for any n > 1.
a(n) = 2 iff n belongs to A000040 (prime numbers).
a(n) = 4 iff n belongs to A001248 (squares of prime numbers).
a(n) = 6 iff n belongs to A006094 (products of two successive prime numbers).
a(n) = 8 iff n belongs to A030078 (cubes of prime numbers).
a(n) = 10 iff n belongs to A090076.
a(n) = 12 iff n belongs to A251720.
a(n) = 14 iff n belongs to A090090.
a(n) = 16 iff n belongs to A030514.
a(n) = 30 iff n belongs to A046301.
a(n) = 32 iff n belongs to A050997.
a(n) = 36 iff n belongs to A166329.
a(1) = 1, for n > 1, a(n) = 2*A246277(n). - Antti Karttunen, Feb 23 2022
a(n) = A122111(A243074(A122111(n))). - Peter Munn, Feb 23 2022
From Peter Munn and Antti Karttunen, May 12 2022: (Start)
a(1) = 1; a(2n) = 2n; a(A003961(n)) = a(n). [complete definition]
a(n) = A005940(1+A322993(n)) = A005940(1+A000265(A156552(n))).
Equivalently, A156552(a(n)) = A000265(A156552(n)).
A297845(a(n), A020639(n)) = n.
A046523(a(n)) = A046523(n).
A071364(a(n)) = A071364(n).
a(n) >= A071364(n).
A243055(a(n)) = A243055(n).
(End)

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

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

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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A348717 a(n) is the least k such that A003961^i(k) = n for some i >= 0 (where A003961^i denotes the i-th iterate of A003961).

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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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A046327 Numbers that are the product of 9 successive primes.

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