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

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

Author

Amarnath Murthy, Oct 23 2002

Keywords

• nonn

Comments

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)

Examples

			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.
		

Links

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

Crossrefs

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

Programs

Maple

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

Mathematica

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

PARI

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

Python

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

Formula

From Michel Marcus, May 29 2022: (Start)

a(n) = A007947(A074167(n)).

a(n) = A007947(A061214(n)). (End)

Extensions

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