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 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. Also numbers m such that m - A007917(m) > A034699(m). - David A. Corneth, Jul 24 2017 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 Cf. A007917, A034699, A081805. Sequence in context: A104395 A267331 A267739 * A254370 A325932 A109024 Adjacent sequences:  A290200 A290201 A290202 * A290204 A290205 A290206 KEYWORD nonn AUTHOR Sílvia Casacuberta Puig, Jul 24 2017 STATUS approved

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Last modified September 18 09:05 EDT 2021. Contains 347518 sequences. (Running on oeis4.)