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
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Sílvia Casacuberta Puig, On the divisibility of binomial coefficients, 2018; see also, arXiv:1906.07652 [math.NT], 2019.
E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, Journal für die reine und angewandte Mathematik, 44:93-146, 1852.
E. Lucas, Théorie des fonctions numériques simplement périodiques, American Journal of Mathematics, 44:184-196, 1878.
J. Shareshian and R. Woodroofe, Divisibility of binomial coefficients and generation of alternating groups, arXiv:1505.05143 [math.CO], 2015-2017.
Wikipedia, Kummer's Theorem
Wikipedia, Lucas' Theorem
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
Sílvia Casacuberta Puig, Jul 24 2017
STATUS
approved