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

%I A076978 #46 May 30 2022 13:00:42
%S A076978 1,2,6,30,6,210,6,2310,2730,30,39270,7410,42,7590,46410,1272810,30,
%T A076978 930930,82110,6,21111090,1230,48969690,1738215570,2310,102,144690,6,
%U A076978 85470,29594505363092670,16770,49990710,138,7849357706190,30,300690390,20223210,1122990,37916970
%N A076978 Product of the distinct primes dividing the product of composite numbers between consecutive primes.
%C A076978 Equivalently, the largest squarefree number that divides the product of composite numbers between successive primes.
%C A076978 From _Robert G. Wilson v_, Dec 02 2020: (Start)
%C A076978 All terms greater than one are even.
%C A076978 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, ..., .
%C A076978 Records: 1, 2, 6, 30, 210, 2310, 2730, 39270, 46410, 1272810, 21111090, ..., (2*A354218).
%C A076978 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, ..., .
%C A076978 (End)
%H A076978 Robert G. Wilson v, <a href="/A076978/b076978.txt">Table of n, a(n) for n = 1..10000</a>
%F A076978 From _Michel Marcus_, May 29 2022: (Start)
%F A076978 a(n) = A007947(A074167(n)).
%F A076978 a(n) = A007947(A061214(n)). (End)
%e A076978 a(4) = product of prime divisors of the product of composite numbers between 7 and 11 = 2 * 3 * 5 = 30.
%e A076978 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.
%p A076978 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
%t A076978 f[n_] := Times @@ (First@# & /@ FactorInteger[Times @@ Range[Prime[n] + 1, Prime[n + 1] - 1]]);  Array[f, 50] (* _Robert G. Wilson v_, Dec 02 2020 *)
%o A076978 (PARI) a(n) = my(p=1); forcomposite(c=prime(n), prime(n+1), p*=c); factorback(factorint(p)[, 1]); \\ _Michel Marcus_, May 29 2022
%o A076978 (Python) from sympy import sieve as p, primefactors
%o A076978 def A076978(n):
%o A076978     result = 1
%o A076978     for composites in range(p[n]+1, p[n+1]):
%o A076978         for primefactor in primefactors(composites):
%o A076978             if result % primefactor != 0: result *= primefactor
%o A076978     return result # _Karl-Heinz Hofmann_, May 30 2022
%Y A076978 Cf. A007947, A061214, A074167, A354217, A354218.
%K A076978 nonn
%O A076978 1,2
%A A076978 _Amarnath Murthy_, Oct 23 2002
%E A076978 More terms from _Emeric Deutsch_, Jul 28 2006
%E A076978 More terms from _Robert G. Wilson v_, Dec 02 2020
%E A076978 Entry revised by _N. J. A. Sloane_, Dec 02 2020