|
|
A290203
|
|
Numbers m having greatest prime power divisor d such that d is smaller than the difference between m and the largest prime smaller than m.
|
|
2
|
|
|
126, 210, 330, 630, 1144, 1360, 2520, 2574, 2992, 3432, 3960, 4199, 4620, 5544, 5610, 5775, 5980, 6006, 6930, 7280, 8008, 8415, 9576, 10005, 10032, 12870, 12880, 13090, 14280, 14586, 15708, 15725, 16182, 17290, 18480, 18837, 19635, 19656, 20475, 20592, 22610
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
It is conjectured that for all integers m there exist two primes p and r such that all the binomial coefficients (m,k) with 1 <= k <= m-1 are divisible by either p or r. Using Lucas's Theorem we can prove that the conjecture is true for integers m such that the difference between m and the largest prime smaller than m is smaller than the greatest prime power divisor of m. Therefore this list examines the numbers m that do not satisfy this property.
|
|
LINKS
|
|
|
EXAMPLE
|
The first number of the sequence is 126. The prime factorization of 126 is 2*3^2*7. Therefore, the greatest prime power divisor is 9. The largest prime smaller than 126 is 113, and then the difference between 126 and 113 is 13. Then 13 is larger than 9 and therefore 126 is part of the sequence.
|
|
MATHEMATICA
|
Reap[For[k = 3, k < 30000, k++, If[k - NextPrime[k, -1] > Max[Power @@@ FactorInteger[k]], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jul 24 2017 *)
|
|
PROG
|
(PARI) isok(n)=my(f = factor(n)); n - precprime(n) > vecmax(vector(#f~, k, f[k, 1]^f[k, 2])); \\ Michel Marcus, Jul 24 2017
(PARI) list(lim)=my(v=List(), p=2, f); forfactored(n=3, lim\1, f=n[2]; if(f[, 2]==[1]~, p=n[1]; next); if(n[1]-p > vecmax(vector(#f~, i, f[i, 1]^f[i, 2])), listput(v, n[1]))); Vec(v) \\ Charles R Greathouse IV, Jul 24 2017
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|