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.

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

Offset: 1

Author

Sílvia Casacuberta Puig, Jul 24 2017

Keywords

• nonn

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.

Also numbers m such that m - A007917(m) > A034699(m). - David A. Corneth, Jul 24 2017

Examples

			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.
		

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

Crossrefs

Cf. A007917, A034699, A081805.

Programs

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 *)

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