login
A079615
Product of all distinct prime factors of all composite numbers between n-th prime and next prime.
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
OFFSET
2,1
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
LINKS
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Reinhard Zumkeller, Jan 29 2003
EXTENSIONS
Corrected by T. D. Noe, May 19 2007
STATUS
approved