A006094 -id:A006094 - cp's OEIS frontend

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

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

A325352 Heinz number of the differences plus one of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 6, 1, 10, 5, 11, 1, 12, 2, 13, 4, 14, 1, 9, 1, 16, 7, 17, 3, 12, 1, 19, 11, 20, 1, 15, 1, 22, 6, 23, 1, 24, 2, 10, 13, 26, 1, 12, 5, 28, 17, 29, 1, 18, 1, 31, 10, 32, 7, 21, 1, 34, 19, 15, 1, 24, 1, 37, 6, 38

Offset: 1

Gus Wiseman, Apr 23 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The only fixed point is 1 because otherwise the sequence decreases omega (A001222) by one.

			The partition (3,2,2,1) with Heinz number 90 has differences plus one (2,1,2) with Heinz number 18, so a(90) = 18.

Positions of m's are A008578 (m = 1), A001248 (m = 2), A006094 (m = 3), A030078 (m = 4), A090076 (m = 5).

Cf. A007294, A049988, A056239, A093641, A112798, A240026, A320466, A325328, A325351, A325360, A325361, A325368, A325405.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
db[n_]:=Times@@Prime/@(1+Differences[primeMS[n]]);
Table[db[n],{n,100}]

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

A037165 a(n) = prime(n)*prime(n+1) - prime(n) - prime(n+1).

Original entry on oeis.org

1, 7, 23, 59, 119, 191, 287, 395, 615, 839, 1079, 1439, 1679, 1931, 2391, 3015, 3479, 3959, 4619, 5039, 5615, 6395, 7215, 8447, 9599, 10199, 10811, 11447, 12095, 14111, 16379, 17679, 18767, 20423, 22199, 23399, 25271, 26891, 28551, 30615, 32039

Offset: 1

Armand Turpel (armandt(AT)unforgettable.com)

a(n) is also the Frobenius number of the numerical semigroup generated by prime(n) and prime(n+1). - Victoria A Sapko (vsapko(AT)math.unl.edu), Feb 21 2001

Joshua Oliver, Table of n, a(n) for n = 1..2000 (first 1000 terms from Vincenzo Librandi)
R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

Frobenius numbers for k successive primes: this sequence (k=2), A138989 (k=3), A138990 (k=4), A138991 (k=5), A138992 (k=6), A138993 (k=7), A138994 (k=8).

Cf. A001043, A006094.

[NthPrime(n)*NthPrime(n+1)-NthPrime(n)-NthPrime(n+1): n in [1..45]]; // Vincenzo Librandi, Dec 18 2012

f[n_] := FrobeniusNumber[{Prime[n], Prime[n + 1]}]; Array[f, 41] (* Robert G. Wilson v, Aug 04 2012 *)
Times@@#-Total[#]&/@Partition[Prime[Range[50]],2,1] (* Harvey P. Dale, Dec 27 2015 *)

a(n)=my(p=prime(n),q=nextprime(p+1)); p*q-p-q \\ Charles R Greathouse IV, Apr 28 2015

a(n) = A006094(n) - A001043(n). - Michel Marcus, Mar 02 2019

A332820 Integers in the multiplicative subgroup of positive rationals generated by the products of two consecutive primes and the cubes of primes. Numbers k for which A048675(k) is a multiple of three.

Original entry on oeis.org

1, 6, 8, 14, 15, 20, 26, 27, 33, 35, 36, 38, 44, 48, 50, 51, 58, 63, 64, 65, 68, 69, 74, 77, 84, 86, 90, 92, 93, 95, 106, 110, 112, 117, 119, 120, 122, 123, 124, 125, 141, 142, 143, 145, 147, 156, 158, 160, 161, 162, 164, 170, 171, 177, 178, 185, 188, 196, 198, 201, 202, 208, 209, 210, 214, 215, 216, 217, 219, 221, 225

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between this sequence, A332821 and A332822, which list the integers in respective cosets of the subgroup.
As the sequence lists the integers in a multiplicative subgroup of the positive rationals, the sequence is closed under multiplication and, provided the result is an integer, under division.
It follows that for any n in this sequence, all powers n^k are present (k >= 0), as are all cubes.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting numbers are a permutation of the full sequence; and if we take the square root of each square term we get the full sequence.
There are no primes in the sequence, therefore if k is present and p is a prime, k*p and k/p are absent (noting that k/p might not be an integer). This property extends from primes to all terms of A050376 (often called Fermi-Dirac primes), therefore to squares of primes, 4th powers of primes etc.
The terms are the even numbers in A332821 halved. The terms are also the numbers m such that 5m is in A332821, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332822, and so on for alternate primes: 7, 13, 19 etc.
The numbers that are half of the even terms of this sequence are in A332822, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332821, which consists exactly of those numbers. These properties extend in a pattern of alternating primes as described in the previous paragraph.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.
If m and n are in this sequence then so is m*n (the definition of "multiplicative semigroup"), while if n is in this sequence, and x is in the complement A359830, then n*x is in A359830. This essentially follows from the fact that A048675 is totally additive sequence. Compare to A329609. - Antti Karttunen, Jan 17 2023

Positions of zeros in A332823; equivalently, numbers in row 3k of A277905 for some k >= 0.
Cf. A048675, A195017, A332821, A332822, A353350 (characteristic function), A353348 (its Dirichlet inverse), A359830 (complement).
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: A000578\{0}, A006094, A090090, A099788, A245630 (A191002 in ascending order), A244726\{0}, A325698, A338471, A338556, A338907.
Subsequence of {1} U A268388.

Select[Range@ 225, Or[Mod[Total@ #, 3] == 0 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]], # == 1] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332820(n) =  { my(f = factor(n)); !((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); };

{a(n) : n >= 1} = {1} U {2 * A332822(k) : k >= 1} U {A003961(a(k)) : k >= 1}.
{a(n) : n >= 1} = {1} U {a(k)^2 : k >= 1} U {A331590(2, A332822(k)) : k >= 1}.
From Peter Munn, Mar 17 2021: (Start)
{a(n) : n >= 1} = {k : k >= 1, 3|A048675(k)}.
{a(n) : n >= 1} = {k : k >= 1, 3|A195017(k)}.
{a(n) : n >= 1} = {A332821(k)/2 : k >= 1, 2|A332821(k)}.
{a(n) : n >= 1} = {A332822(k)/3 : k >= 1, 3|A332822(k)}.
(End)

New name from Peter Munn, Mar 08 2021

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

A104210 Positive integers divisible by at least 2 consecutive primes.

Original entry on oeis.org

6, 12, 15, 18, 24, 30, 35, 36, 42, 45, 48, 54, 60, 66, 70, 72, 75, 77, 78, 84, 90, 96, 102, 105, 108, 114, 120, 126, 132, 135, 138, 140, 143, 144, 150, 154, 156, 162, 165, 168, 174, 175, 180, 186, 192, 195, 198, 204, 210, 216, 221, 222, 225, 228, 231, 234, 240

Offset: 1

Leroy Quet, Mar 13 2005

nonn

If a perfect square is in this sequence, then so is its square root (e.g., 144 and 12). - Alonso del Arte, May 07 2012

The numbers of terms not exceeding 10^k, for k=1,2,..., are 1, 22, 242, 2456, 24632, 246414, 2464272, 24643281, 246433426, ... Apparently, the asymptotic density of this sequence is 0.24643... - Amiram Eldar, Apr 10 2021

			35 is divisible by both 5 and 7, and 5 and 7 are consecutive primes.
77 is divisible by both 7 and 11, and 7 and 11 are consecutive primes.
110 is not in the sequence because, although it is divisible by 2, 5 and 11, it is not divisible by 3 or 7.

Cf. A003961, A296210 (characteristic function), A319630 (complement), A379230 [= A252748(a(n))].
Positions of terms larger than 1 in A300820 and in A322361.
Subsequences: A006094, A349169 (conjectured, after its initial 1), A349176, A355527 (squarefree terms), A372566, A378884, A379232.

N:= 1000: # for terms <= N
R:= {}:
p:= 2:
do
  q:= p; p:= nextprime(p);
  if p*q > N then break fi;
  R:= R union {seq(i,i=p*q..N,p*q)}
od:
sort(convert(R,list)); # Robert Israel, Apr 13 2020

fQ[n_] := Block[{lst = PrimePi /@ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]}, Count[ Drop[lst, 1] - Drop[lst, -1], 1] > 0]; Select[ Range[244], fQ[ # ] &] (* Robert G. Wilson v, Mar 16 2005 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
is_A104210(n) = (gcd(n,A003961(n))>1); \\ Antti Karttunen, Dec 24 2024

{k such that gcd(k, A003961(k)) > 1}. - Antti Karttunen, Dec 24 2024

More terms from Robert G. Wilson v, Mar 16 2005

A090076 a(n) = prime(n)*prime(n+2).

Original entry on oeis.org

10, 21, 55, 91, 187, 247, 391, 551, 713, 1073, 1271, 1591, 1927, 2279, 2773, 3233, 3953, 4331, 4891, 5609, 6059, 7031, 8051, 8989, 9991, 10807, 11227, 12091, 13843, 14803, 17399, 18209, 20413, 20989, 23393, 24613, 26219, 28199, 29893, 31313

Offset: 1

Felix Tubiana, Jan 21 2004

Subsequence of A192133. - Reinhard Zumkeller, Jun 26 2011

For n > 1: A078898(a(n)) = 4. - Reinhard Zumkeller, Apr 06 2015

			a(5) = prime(5)*prime(7) = 11*17 = 187.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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. A006094, A090090.

Cf. A078898.

a090076 n = a090076_list !! (n-1)
a090076_list = zipWith (*) a000040_list $ drop 2 a000040_list
-- Reinhard Zumkeller, Dec 17 2014

Table[Prime[n] Prime[n + 2], {n, 1, 40}] (* Robert G. Wilson v, Jan 22 2004 *)
Last[#]First[#]&/@Partition[Prime[Range[50]],3,1] (* Harvey P. Dale, May 08 2013 *)

ithprime(i)*ithprime(i+2) $ i = 1..40 // Zerinvary Lajos, Feb 26 2007

def prime_gaps(n):
    primegaps = []
    nprimes = primes_first_n(n+1)
    for i in range(2, n+1):
        primegaps.append(nprimes[i]*nprimes[i-2])
    return primegaps
print(prime_gaps(60)) # Zerinvary Lajos, Jul 08 2008

Extended by Robert G. Wilson v, Jan 22 2004

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

A089581 a(n) = prime(2n-1)prime(2*n).

Original entry on oeis.org

6, 35, 143, 323, 667, 1147, 1763, 2491, 3599, 4757, 5767, 7387, 9797, 11021, 12317, 16637, 19043, 22499, 25591, 28891, 32399, 36863, 39203, 47053, 51983, 55687, 60491, 67591, 72899, 77837, 82919, 95477, 99221, 111547, 121103, 126727, 136891

Offset: 1

Norbert A'Campo (nac(AT)member.ams.org), Dec 29 2003

a(n+1) is the smallest squarefree composite number that is relatively prime to the first 2*n prime numbers.

Bisection of A006094. - R. J. Mathar, Jan 23 2013

			a(13)=9797

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Table[Prime[2 n - 1] Prime[2 n], {n, 30}] (* Michael De Vlieger, Jul 21 2016 *)
Times@@@Partition[Prime[Range[80]],2] (* Harvey P. Dale, Jan 12 2022 *)

p=2;q=3;i=1;a=vector(100);for(i=1,100,a[i]=p*q;p=nextprime(q+1);q=nextprime(p+1));a

prime(2*n-1)*prime(2*n), n=1, 2, 3 ...

Edited by Zak Seidov, Jun 13 2006

Typo in PARI program fixed by Zak Seidov, May 19 2010

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A325352 Heinz number of the differences plus one of the integer partition with Heinz number n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A046301 Product of 3 successive primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

Extensions

A037165 a(n) = prime(n)*prime(n+1) - prime(n) - prime(n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A332820 Integers in the multiplicative subgroup of positive rationals generated by the products of two consecutive primes and the cubes of primes. Numbers k for which A048675(k) is a multiple of three.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A046303 Product of 5 successive primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A104210 Positive integers divisible by at least 2 consecutive primes.

Views

Author

Keywords

Comments

Examples

A089581 a(n) = prime(2n-1)prime(2*n).