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. (Formerly M2515)

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

1,1

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, and H. C. Williams, Further Tabulation of the Erdős-Selfridge Function, Math. Comput. 66 (1997) 1709-1717.

R. Scheidler and 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, and 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.

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}]
(* Alternative: *)
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 *)

Prog

(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 */
(PARI) primorial(x)=prod(i=1, primepi(x), prime(i))
a(n)=my(G=primorial(n), N=n+1); while(gcd(binomial(N++, n), G)!=1, ); N \\ Charles R Greathouse IV, Apr 28 2026

Crossrefs

Sequence in context: A251532 A251533 A295849 * A133339 A112267 A068985

Adjacent sequences: A003455 A003456 A003457 * A003459 A003460 A003461

Keyword

nonn,nice

Author

N. J. A. Sloane, Simon Plouffe

Extensions

Extended by Robert G. Wilson v, Dec 01 2002

Status

approved