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

1,2

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)

Links

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

Formula

From Michel Marcus, May 29 2022: (Start)

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

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A354418 A074168 A079615 * A117213 A127797 A338441

Adjacent sequences: A076975 A076976 A076977 * A076979 A076980 A076981

Keyword

nonn

Author

Amarnath Murthy, Oct 23 2002

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

Status

approved