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%I #44 Jun 26 2019 09:30:26
%S 126,210,330,630,1144,1360,2520,2574,2992,3432,3960,4199,4620,5544,
%T 5610,5775,5980,6006,6930,7280,8008,8415,9576,10005,10032,12870,12880,
%U 13090,14280,14586,15708,15725,16182,17290,18480,18837,19635,19656,20475,20592,22610
%N 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.
%C 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.
%C Also numbers m such that m - A007917(m) > A034699(m). - _David A. Corneth_, Jul 24 2017
%H Charles R Greathouse IV, <a href="/A290203/b290203.txt">Table of n, a(n) for n = 1..10000</a>
%H Sílvia Casacuberta Puig, <a href="https://math.mit.edu/research/highschool/rsi/documents/2017Puig.pdf">On the divisibility of binomial coefficients</a>, 2018; see <a href="https://arxiv.org/abs/1906.07652">also</a>, arXiv:1906.07652 [math.NT], 2019.
%H E. Kummer, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002147432">Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen</a>, Journal für die reine und angewandte Mathematik, 44:93-146, 1852.
%H E. Lucas, <a href="http://www.jstor.org/stable/2369308">Théorie des fonctions numériques simplement périodiques</a>, American Journal of Mathematics, 44:184-196, 1878.
%H J. Shareshian and R. Woodroofe, <a href="https://arxiv.org/abs/1505.05143">Divisibility of binomial coefficients and generation of alternating groups</a>, arXiv:1505.05143 [math.CO], 2015-2017.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Kummer%27s_theorem">Kummer's Theorem</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lucas%27s_theorem">Lucas' Theorem</a>
%e 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.
%t 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 *)
%o (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
%o (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
%Y Cf. A007917, A034699, A081805.
%K nonn
%O 1,1
%A _Sílvia Casacuberta Puig_, Jul 24 2017
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