A079615 Product of all distinct prime factors of all composite numbers between n-th prime and next prime.

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

Offset: 2

Author

Reinhard Zumkeller, Jan 29 2003

Keywords

• nonn
• nice

Comments

a(n) = A007947(A056831(n)), squarefree kernel of least common multiple of composite numbers between n-th prime and next prime.

Note that each term is a product of distinct primes. - T. D. Noe, May 19 2007

Equals A076978 without its first term. - R. J. Mathar, Sep 19 2008

Same for A074168. - Georg Fischer, Oct 06 2018

For n > 2, a(n) is of the form 2*3*r, where r is relatively prime to 6. Therefore, for every n > 2, a(n) is a Zumkeller number (see Corollary 5, Rao/Peng link). - Ivan N. Ianakiev, Jan 24 2020

Examples

			n=9: factorizations of numbers between 23=A000040(9) and 29=A000040(10) are 24=3*2^3, 25=5^2, 26=13*2 and 27=3^3, therefore a(9) = 2*3*5*7*13 = 2730.
		

Links

• T. D. Noe, Table of n, a(n) for n = 2..1000
• K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.

Crossrefs

Cf. A005117, A002110, A083207.

Programs

Mathematica

a[n_] := (p = Prime[n]; s = Select[Table[k, {k, p, NextPrime[p]}], ! PrimeQ[#] &]; Times @@ ((FactorInteger /@ s // Flatten[#, 1] &)[[All, 1]] // Union)); a /@ Range[2, 35] (* Jean-François Alcover, Jul 13 2011 *)
Table[Times@@Union[Flatten[Transpose[FactorInteger[#]][[1]]&/@ (Range[ Prime[ n]+1, NextPrime[Prime[n]]-1])]],{n,2,50}] (* Harvey P. Dale, Oct 10 2011 *)

Extensions

Corrected by T. D. Noe, May 19 2007