|
|
A308078
|
|
Numbers k such that binomial(k^2,k) - k^k is squarefree.
|
|
1
|
|
|
2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 77, 79
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
The sequence appears to contain most primes (except 11, ...) and some odd semiprimes (33, 65, 77, ...). What can be said about these "exceptional" values? What are the first terms with more prime factors?
The sequence remains nearly the same if k^k is replaced by k^2, cf. A309290. Then 0 and 11 are in the sequence but 3, 13, 37 and 53 are not.
97 is a term (see SeqFan list discussion). - Chai Wah Wu, Jul 20 2020
|
|
LINKS
|
|
|
MATHEMATICA
|
Select[Range[0, 50], SquareFreeQ[Binomial[#^2, #] - #^#] &]
|
|
PROG
|
(PARI) is(n)=issquarefree(binomial(n^2, n)-n^n)
for(n=0, oo, is(n) && print1(n, ", "))
(Magma) [0] cat [n: n in [2..45] | IsSquarefree(Binomial(n^2, n) - n^n)];
|
|
CROSSREFS
|
Cf. A309289 (binomial(2k,k) - k^2 is prime), A309290 (binomial(k^2,k) - k^2 is squarefree).
|
|
KEYWORD
|
nonn,hard,more
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|