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A003458 Erdős-Selfridge function: a(n) is the least number m > n+1 such that the least prime factor of binomial(m, n) is > n.

%I A003458 M2515 #45 Feb 16 2025 08:32:27
%S A003458 3,6,7,7,23,62,143,44,159,46,47,174,2239,239,719,241,5849,2098,2099,
%T A003458 43196,14871,19574,35423,193049,2105,36287,1119,284,240479,58782,
%U A003458 341087,371942,6459,69614,37619,152188,152189,487343,767919,85741,3017321
%N A003458 Erdős-Selfridge function: a(n) is the least number m > n+1 such that the least prime factor of binomial(m, n) is > n.
%D A003458 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003458 Jonathan Webster, <a href="/A003458/b003458.txt">Table of n, a(n) for n = 1..375</a> (terms 1..200 from H. C. Williams)
%H A003458 E. F. Ecklund, Jr. et al., <a href="http://dx.doi.org/10.1090/S0025-5718-1974-0337732-2">A new function associated with the prime factors of C(n,k)</a>, Math. Comp., 28 (1974), 647-649.
%H A003458 R. F. Lukes; R. Scheidler; H. C. Williams, <a href="http://dx.doi.org/10.1090/S0025-5718-97-00864-8">Further Tabulation of the Erdos-Selfridge Function</a>, Math. Comput. 66 (1997) 1709-1717.
%H A003458 R. Scheidler, H. C. Williams, <a href="http://dx.doi.org/10.1090/S0025-5718-1992-1134737-X">A method for tabulating the number-theoretic function g(k)</a>, Math. Comp. 59 (199) (1992) 251-257.
%H A003458 Brianna Sorenson, Jonathan P Sorenson, Jonathan Webster, <a href="https://arxiv.org/abs/1907.08559">An Algorithm and Estimates for the Erdős-Selfridge Function (work in progress)</a>, arXiv:1907.08559 [math.NT], 2019.
%H A003458 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Erdos-SelfridgeFunction.html">Erdős-Selfridge function.</a>
%p A003458 A003458 := proc(n)
%p A003458     local m;
%p A003458     for m from n+2 do
%p A003458         if A020639( binomial(m,n)) > n then
%p A003458             return m ;
%p A003458         end if;
%p A003458     end do:
%p A003458 end proc:
%p A003458 seq(A003458(n),n=1..16) ; # _R. J. Mathar_, Mar 27 2024
%t A003458 f[n_] := Block[{k = n + 2, p = Table[Prime[i], {i, 1, PrimePi[n]}]}, While[ First[ Sort[ Mod[ Binomial[k, n], p]]] == 0, k++ ]; k]; Table[ f[n], {n, 1, 40}]
%t A003458 esf[n_]:=Module[{m=n+2},While[FactorInteger[Binomial[m,n]][[1,1]]<=n, m++];m]; Array[esf,50] (* _Harvey P. Dale_, Nov 03 2013 *)
%o A003458 (PARI) a(n) = local(m,i,f); m=0; i=n+1; while(m<=n,i=i+1; m=factor(binomial(i,n))[1,1]); i /* _Ralf Stephan_ */
%K A003458 easy,nonn,nice
%O A003458 1,1
%A A003458 _N. J. A. Sloane_, _Simon Plouffe_
%E A003458 Extended by _Robert G. Wilson v_, Dec 01 2002