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.

3, 6, 7, 7, 23, 62, 143, 44, 159, 46, 47, 174, 2239, 239, 719, 241, 5849, 2098, 2099, 43196, 14871, 19574, 35423, 193049, 2105, 36287, 1119, 284, 240479, 58782, 341087, 371942, 6459, 69614, 37619, 152188, 152189, 487343, 767919, 85741, 3017321

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• easy
• nonn
• nice

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Jonathan Webster, Table of n, a(n) for n = 1..375 (terms 1..200 from H. C. Williams)
• E. F. Ecklund, Jr. et al., A new function associated with the prime factors of C(n,k), Math. Comp., 28 (1974), 647-649.
• R. F. Lukes; R. Scheidler; H. C. Williams, Further Tabulation of the Erdos-Selfridge Function, Math. Comput. 66 (1997) 1709-1717.
• R. Scheidler, H. C. Williams, A method for tabulating the number-theoretic function g(k), Math. Comp. 59 (199) (1992) 251-257.
• Brianna Sorenson, Jonathan P Sorenson, Jonathan Webster, An Algorithm and Estimates for the Erdős-Selfridge Function (work in progress), arXiv:1907.08559 [math.NT], 2019.
• Eric Weisstein's World of Mathematics, Erdős-Selfridge function.

Programs

Maple

A003458 := proc(n)
    local m;
    for m from n+2 do
        if A020639( binomial(m,n)) > n then
            return m ;
        end if;
    end do:
end proc:
seq(A003458(n),n=1..16) ; # R. J. Mathar, Mar 27 2024

Mathematica

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}]
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 *)

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 */

Extensions

Extended by Robert G. Wilson v, Dec 01 2002