A361761 -id:A361761 - cp's OEIS frontend

A061214 Product of composite numbers between the n-th and (n+1)st primes.

1, 4, 6, 720, 12, 3360, 18, 9240, 11793600, 30, 45239040, 59280, 42, 91080, 311875200, 549853920, 60, 1072431360, 328440, 72, 2533330800, 531360, 4701090240, 60072730099200, 970200, 102, 1157520, 108, 1367520, 1063186156509747740870400000, 2146560, 43191973440

Offset: 1

Amarnath Murthy, Apr 21 2001

nonn

			a(4) = 8 * 9 * 10 = 720. 7 is the fourth prime and 11 is the fifth prime. a(5) = 12 as 11 and 13 both are primes.

T. D. Noe, Table of n, a(n) for n = 1..2000

Cf. A006530, A052248, A052297, A072472, A361760, A361761.

Cf. A046933 and A054265 (number and sum of these composites).

a061214 n = a061214_list !! (n-1)
a061214_list = f a000040_list where
   f (p:ps'@(p':ps)) = (product [p+1..p'-1]) : f ps'
-- Reinhard Zumkeller, Jun 22 2011

A061214 := proc(n)
    local k ;
    product(k,k=ithprime(n)+1..ithprime(n+1)-1) ;
end proc: # R. J. Mathar, Apr 23 2013

Table[Times@@Range[Prime[n]+1,Prime[n+1]-1],{n,30}] (* Harvey P. Dale, Jun 14 2011 *)
Times@@Range[#[[1]]+1,#[[2]]-1]&/@Partition[Prime[Range[40]],2,1] (* Harvey P. Dale, Apr 23 2022 *)

{ n=0; q=2; forprime (p=3, prime(2001), a=1; for (i=q + 1, p - 1, a*=i); q=p; write("b061214.txt", n++, " ", a) ) } \\ Harry J. Smith, Jul 19 2009

v=primes(100);for(i=1,#v-1,v[i]=prod(j=v[i]+1,v[i+1]-1,j));vecextract(v,"1..-2") \\ Charles R Greathouse IV, Feb 27 2012

from math import prod
from sympy import prime
def A061214(n): return prod(i for i in range(prime(n)+1,prime(n+1))) # Chai Wah Wu, Jul 10 2022

A006530(a(n)) = A052248(n) for n > 1. - Reinhard Zumkeller, Jun 22 2011

More terms from James Sellers, Apr 24 2001

Better definition from T. D. Noe, Jan 21 2008

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

A061214 Product of composite numbers between the n-th and (n+1)st primes.

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

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

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