A061214 -id:A061214 - cp's OEIS frontend

A229992 Numbers for which A061214(n) + 1 is prime, where A061214(n) = product of composite numbers between prime(n) and prime(n+1) .

2, 3, 5, 6, 7, 8, 10, 12, 13, 14, 16, 17, 20, 21, 25, 26, 28, 29, 31, 33, 35, 39, 41, 43, 44, 45, 49, 51, 52, 57, 60, 64, 67, 69, 70, 81, 83, 85, 89, 90, 91, 97, 98, 104, 109, 113, 116, 118, 120, 131, 134, 136, 140, 142, 144, 145, 148, 152, 157, 171, 173

Offset: 2

Clark Kimberling, Oct 09 2013

nonn

			a(2) = 2 because 4 + 1 is prime; a(3) = 3 because 6 + 1 is prime; 4 is not in A229992 because 8*9*10 + 1 is not prime.

Cf. A061214, A229992.

q[n_] := Product[k, {k, Prime[n] + 1, Prime[n + 1] - 1}]; c[n_] := If[PrimeQ[q[n] + 1], 1, 0]; t = Table[c[n], {n, 1, 230}]; u = Rest[Flatten[Position[t, 1]]]

A229991 Numbers for which A061214(n) - 1 is prime, where A061214(n) = product of composite numbers between prime(n) and prime(n+1) .

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 19, 20, 22, 26, 28, 29, 33, 34, 35, 41, 43, 45, 49, 52, 55, 56, 57, 60, 61, 64, 69, 72, 75, 81, 83, 85, 86, 89, 90, 91, 93, 94, 98, 104, 105, 109, 113, 116, 120, 122, 123, 124, 129, 134, 138, 139, 140, 142, 143

Offset: 2

Clark Kimberling, Oct 09 2013

nonn

			a(2) = 2 because 4 - 1 is prime; a(4) = 4 because 8*9*10-1 is prime; 12 is not in A229991 because 38*39*40 - 1 is not prime.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A061214, A229992.

q[n_] := Product[k, {k, Prime[n] + 1, Prime[n + 1] - 1}]; c[n_] := If[PrimeQ[q[n] - 1], 1, 0]; t = Table[c[n], {n, 1, 230}]; u = Flatten[Position[t, 1]]
Select[Range[150],PrimeQ[Times@@Range[Prime[#]+1,Prime[#+1]-1]-1]&] (* Harvey P. Dale, Feb 15 2014 *)

A229993 Numbers for which c(n) - 1 and c(n) + 1 are twin primes, where c(n) = A061214(n) = product of composite numbers between prime(n) and prime(n+1) .

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 13, 14, 17, 20, 26, 28, 29, 33, 35, 41, 43, 45, 49, 52, 57, 60, 64, 69, 81, 83, 85, 89, 90, 91, 98, 104, 109, 113, 116, 120, 134, 140, 142, 144, 148, 152, 171, 173, 176, 178, 182, 190, 201, 202, 204, 206, 209, 212, 215, 225, 230, 234

Offset: 2

Clark Kimberling, Oct 09 2013

nonn

			c(n) - 1:  3, 5, 719, 11, 3359, 17, 9239.
c(n) + 1: 5, 7, 721, 13, 3361, 19, 9241.  Here, for example, for we have twin primes except for n = 4, since 721 is not prime.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A061214, A229991, A229992.

z = 400; c[n_] := Product[k, {k, Prime[n] + 1, Prime[n + 1] - 1}]; d[n_] := If[PrimeQ[c[n] - 1], 1, 0]; t1 = Table[d[n], {n, 1, z}]; u1 = Flatten[Position[t1, 1]]; e[n_] := If[PrimeQ[c[n] + 1], 1, 0]; t2 = Table[e[n], {n, 1, z}]; u2 = Flatten[Position[t2, 1]]; u = Intersection[u1, u2]
pcnQ[n_]:=Module[{p=Times@@Range[Prime[n]+1,Prime[n+1]-1]},AllTrue[p+{1,-1},PrimeQ]]; Select[Range[250],pcnQ] (* Harvey P. Dale, Jan 28 2023 *)

A219611 a(n) is the smallest omega(A061214(k)) sampled over all indices k of prime gaps prime(k+1) - prime(k) = 2n, where omega = A001221.

Original entry on oeis.org

1, 3, 5, 9, 11, 14, 14, 21

Offset: 1

Naohiro Nomoto, Apr 12 2013

The example demonstrates that the minimum order of the set of primes represented by all composites in the prime gap 2*n is not necessarily obtained by using the smallest prime(k) (that would be A038664).

			For n=8: p_283-p_282 = p_296-p_295 = 2*8=16; omega(A061214(282)) > omega(A061214(295)); omega(A061214(295)) = 21; so a(8) = 21.

Cf. A052297.

A052248 Greatest prime divisor of all composite numbers between p and next prime.

Original entry on oeis.org

2, 3, 5, 3, 7, 3, 11, 13, 5, 17, 19, 7, 23, 17, 29, 5, 31, 23, 3, 37, 41, 43, 47, 11, 17, 53, 3, 37, 61, 43, 67, 23, 73, 5, 31, 79, 83, 43, 89, 5, 61, 3, 97, 11, 103, 109, 113, 19, 29, 79, 5, 83, 127, 131, 89, 5, 137, 139, 47, 97, 151, 103, 13, 157, 163, 167, 173, 29, 13

Offset: 2

N. J. A. Sloane

Or, largest of all prime factors of the numbers between prime(n) and prime(n+1).
a(n) = 3, 5, 7, 11, 13 iff prime(n) is in A059960, A080185, A080186, A080187, A080188 respectively. This sequence defines a mapping f of primes > 2 to primes (cf. A080189) and f(p) < p holds for all p > 2. - Klaus Brockhaus, Feb 10 2003
a(n) = A006530(A061214(n)). - Reinhard Zumkeller, Jun 22 2011

			a(8) = 11 since 20 = 2*2*5, 21 = 3*7, 22 = 2*11 are the numbers between prime(8) = 19 and prime(9) = 23.
For n=9, n-th prime is 23, composites between 23 and next prime are 24 25 26 27 29 of which largest prime divisor is 13, so a(9)=13.

T. D. Noe, Table of n, a(n) for n = 2..1000

Cf. A006530, A059960, A080185, A080186, A080187, A080188, A080189, A052180, A065091.

a052248 n = a052248_list !! (n-2)
a052248_list = f a065091_list where
   f (p:ps'@(p':ps)) = (maximum $ map a006530 [p+1..p'-1]) : f ps'
-- Reinhard Zumkeller, Jun 22 2011

g[n_] := Block[{t = Range[Prime[n] + 1, Prime[n + 1] - 1]}, Max[First /@ Flatten[ FactorInteger@t, 1]]]; Table[ g[n], {n, 2, 72}] (* Robert G. Wilson v, Feb 08 2006 *)
cmp[{a_,b_}]:=Max[Flatten[FactorInteger/@Range[a+1,b-1],1][[All,1]]]; cmp/@ Partition[ Prime[Range[2,80]],2,1] (* Harvey P. Dale, May 16 2020 *)

forprime(p=3,360,q=nextprime(p+1); m=0; for(j=p+1,q-1,f=factor(j); a=f[matsize(f)[1],1]; if(m

a(n) = max(prime(n) < k < prime(n+1), A006530(k)).

A052297 Number of distinct prime factors of all composite numbers between n-th and (n+1)st primes.

Original entry on oeis.org

0, 1, 2, 3, 2, 4, 2, 5, 5, 3, 6, 5, 3, 5, 6, 7, 3, 7, 6, 2, 8, 4, 8, 9, 5, 3, 6, 2, 6, 14, 5, 8, 3, 11, 3, 9, 7, 6, 8, 8, 3, 13, 2, 6, 3, 14, 15, 5, 3, 7, 9, 3, 11, 8, 9, 9, 3, 9, 6, 3, 13, 16, 7, 3, 6, 16, 8, 13, 3, 6, 9, 10, 9, 9, 6, 8, 11, 6, 12, 14, 4, 14, 2, 10, 7, 8, 11, 6, 4, 6, 16, 10, 6, 13

Offset: 1

Labos Elemer, Feb 09 2000

nonn

From Lei Zhou, Mar 18 2014: (Start)
This is also the number of primes such that the (n+1)-th prime (mod i-th prime) is smaller than the (n+1)-th prime (mod n-th prime) for 1 <= i < n.
Proof: We denote the n-th prime number as P_n. Suppose P_(n+1) mod P_i = k; we can write P_(n+1) = m*P_i + k. Setting l = P_(n+1) - P_n, the composite numbers between P_n and P_(n+1) will be consecutively m*P_i + C, where C = k-l+1, k-l+2, ..., k-1. If k < l, there must be a value at which C equals zero since k-1 > 0 and k-l+1 <= 0, so P_i is a factor of a composite number between P_n and P_(n+1). If k >= l, all C values are greater than zero, thus P_i cannot be a factor of a composite number between P_n and P_(n+1). (End)

			n=30, p(30)=113, the next prime is 127. Between them are 13 composites: {114, 115, ..., 126}. Factorizing all and collecting prime factors, the set {2,3,5,7,11,13,17,19,23,29,31,41,59,61} is obtained, consisting of 14 primes, so a(30)=14.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A052180, A052248, A061214, A077218.

Length[Union[Flatten[Table[Transpose[FactorInteger[n]][[1]],{n, First[#]+ 1, Last[#]-1}]]]]&/@Partition[Prime[Range[100]],2,1] (* Harvey P. Dale, Jan 19 2012 *)

A076978 Product of the distinct primes dividing the product of composite numbers between consecutive primes.

Original entry on oeis.org

1, 2, 6, 30, 6, 210, 6, 2310, 2730, 30, 39270, 7410, 42, 7590, 46410, 1272810, 30, 930930, 82110, 6, 21111090, 1230, 48969690, 1738215570, 2310, 102, 144690, 6, 85470, 29594505363092670, 16770, 49990710, 138, 7849357706190, 30, 300690390, 20223210, 1122990, 37916970

Offset: 1

Amarnath Murthy, Oct 23 2002

nonn

Equivalently, the largest squarefree number that divides the product of composite numbers between successive primes.
From Robert G. Wilson v, Dec 02 2020: (Start)
All terms greater than one are even.
Omega(a(n)): 0, 1, 2, 3, 2, 4, 2, 5, 5, 3, 6, 5, 3, 5, 6, 7, 3, 7, 6, 2, 8, 4, 8, 9, 5, ..., .
Records: 1, 2, 6, 30, 210, 2310, 2730, 39270, 46410, 1272810, 21111090, ..., (2*A354218).
Factored: 1, 2, 2*3, 2*3*5, 2*3*5*7, 2*3*5*7*11, 2*3*5*7*13, 2*3*5*7*11*17, 2*3*5*7*13*17, 2*3*5*7*11*19*29, ..., .
(End)

			a(4) = product of prime divisors of the product of composite numbers between 7 and 11 = 2 * 3 * 5 = 30.
a(5)=6 because 12 is the only composite number between the 5th and the 6th primes (11 and 13) and largest squarefree divisor of 12 is 6.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A007947, A061214, A074167, A354217, A354218.

with(numtheory): b:=proc(j) if issqrfree(j) then j else fi end: a:=proc(n) local B,BB: B:=divisors(product(i,i=ithprime(n)+1..ithprime(n+1)-1)): BB:=(seq(b(B[j]),j=1..nops(B))): max(BB); end: seq(a(n),n=1..33); # Emeric Deutsch, Jul 28 2006

f[n_] := Times @@ (First@# & /@ FactorInteger[Times @@ Range[Prime[n] + 1, Prime[n + 1] - 1]]);  Array[f, 50] (* Robert G. Wilson v, Dec 02 2020 *)

a(n) = my(p=1); forcomposite(c=prime(n), prime(n+1), p*=c); factorback(factorint(p)[, 1]); \\ Michel Marcus, May 29 2022

from sympy import sieve as p, primefactors
def A076978(n):
    result = 1
    for composites in range(p[n]+1, p[n+1]):
        for primefactor in primefactors(composites):
            if result % primefactor != 0: result *= primefactor
    return result # Karl-Heinz Hofmann, May 30 2022

From Michel Marcus, May 29 2022: (Start)
a(n) = A007947(A074167(n)).
a(n) = A007947(A061214(n)). (End)

More terms from Emeric Deutsch, Jul 28 2006
More terms from Robert G. Wilson v, Dec 02 2020
Entry revised by N. J. A. Sloane, Dec 02 2020

A092435 Prime factorials divided by their corresponding primorials.

Original entry on oeis.org

1, 1, 4, 24, 17280, 207360, 696729600, 12541132800, 115880067072000, 1366643159020339200000, 40999294770610176000000, 1854768736099424576471040000000, 109950690675973888893203251200000000, 4617929008390903333514536550400000000, 420600974084243475616503989010432000000000

Offset: 1

Don Willard (dwillard(AT)prairie.cc.il.us), Mar 23 2004

nonn

			E.g., 2 factorial divided by 2 primorial is 1; 3 factorial is 6, divided by 3 primorial (3*2=6) is also 1; 5 factorial is 120, divided by 5 primorial (5*3*2=30) is 4 and so forth.

Subsequence of A036691. - Chayim Lowen, Jul 23 2015

Cf. A002110, A039716.

a:= proc(n) option remember; `if`(n<2, 1,
      a(n-1)*mul(i, i=ithprime(n-1)+1..ithprime(n)-1))
    end:
seq(a(n), n=1..15);  # Alois P. Heinz, Jan 15 2025

Table[ Prime[n]! / Times @@ Prime[ Range[ n]], {n, 13}] (* Robert G. Wilson v, Mar 25 2004 *)

a(n)=prime(n)!/prod(i=1,n,prime(i)) \\ Ralf Stephan, Dec 21 2013

p!/p# = A039716/A002110.
Partial products of A061214. - Lekraj Beedassy, Nov 06 2006
From Chayim Lowen, Jul 23 - Aug 05 2015: (Start)
a(n) = A036691(A065890(n)).
a(n) = A000142(A002808(A065890(n)))/A034386(A002808(A065890(n))).
a(n) = Product_{k=1..n} prime(k)^(A085604(prime(n),k)-1).
a(n) = A049614(prime(n)).
a(n) = Product_{k=1..prime(n)} k^A066247(k). (End)

Edited by Robert G. Wilson v, Mar 25 2004

More terms from Michel Marcus, Jan 15 2025

A072472 a(n) = product of numbers from prime(n)+1 up to prime(n+1), where prime(n) is the n-th prime.

Original entry on oeis.org

3, 20, 42, 7920, 156, 57120, 342, 212520, 342014400, 930, 1673844480, 2430480, 1806, 4280760, 16529385600, 32441381280, 3660, 71852901120, 23319240, 5256, 200133133200, 44102880, 418397031360, 5827054819622400, 97990200, 10506, 123854640, 11772, 154529760

Offset: 1

Amarnath Murthy, Jun 20 2002

nonn

Originally the offset was -1 and two terms more in front were pre defined in a inscrutable manner (a(-1)=1, a(0) = 2; .....). - Karl-Heinz Hofmann, Apr 18 2023

			a(1) = 3
a(2) = 4 * 5 = 20
a(3) = 6 * 7 = 42
a(4) = 8 * 9 * 10 * 11 = 7920

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A000040, A061214, A361760, A361761.

a[n_] := Product[k, {k, Prime[n] + 1, Prime[n + 1]}]; Table[a[n], {n, 1, 30}]

a(n) = prod(k=prime(n)+1, prime(n+1), k); \\ Michel Marcus, Apr 17 2023

from sympy import prod, sieve as prime
def A072472(n):
    return prod(range(prime[n]+1, prime[n+1]+1)) # Karl-Heinz Hofmann, Apr 17 2023

a(n) = A000040(n+1)*A061214(n). - Karl-Heinz Hofmann, Apr 18 2023

Edited by Robert G. Wilson v, Jun 21 2002

Offset, data and name corrected by Karl-Heinz Hofmann, Apr 18 2023

A361760 a(n) = Product_{i=prime(n)..prime(n+1)-1} i.

Original entry on oeis.org

2, 12, 30, 5040, 132, 43680, 306, 175560, 271252800, 870, 1402410240, 2193360, 1722, 3916440, 14658134400, 29142257760, 3540, 65418312960, 22005480, 5112, 184933148400, 41977440, 390190489920, 5346472978828800, 94109400, 10302, 119224560, 11556, 149059680, 120140035685601494718355200000, 272613120

Offset: 1

Karl-Heinz Hofmann, Mar 23 2023

nonn

Winston de Greef, Table of n, a(n) for n = 1..10000

Cf. A061214, A072472, A361761.

a(n) = my(x=1); for(i=prime(n), prime(n+1)-1, x*=i); x; \\ Michel Marcus, Mar 28 2023

from sympy import prod, sieve
def A361760(n): return prod(range(sieve[n], sieve[n+1]))

a(n) = A000040(n)*A061214(n).

cp's OEIS Frontend

A229992 Numbers for which A061214(n) + 1 is prime, where A061214(n) = product of composite numbers between prime(n) and prime(n+1) .

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A229991 Numbers for which A061214(n) - 1 is prime, where A061214(n) = product of composite numbers between prime(n) and prime(n+1) .

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A229993 Numbers for which c(n) - 1 and c(n) + 1 are twin primes, where c(n) = A061214(n) = product of composite numbers between prime(n) and prime(n+1) .

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A219611 a(n) is the smallest omega(A061214(k)) sampled over all indices k of prime gaps prime(k+1) - prime(k) = 2n, where omega = A001221.

Views

Author

Keywords

Comments

Examples

Crossrefs

A052248 Greatest prime divisor of all composite numbers between p and next prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A052297 Number of distinct prime factors of all composite numbers between n-th and (n+1)st primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A076978 Product of the distinct primes dividing the product of composite numbers between consecutive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A092435 Prime factorials divided by their corresponding primorials.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple