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Erdős-Selfridge Function

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The Erdős-Selfridge function g(k) is defined as the least integer bigger than k+1 such that the least prime factor of (g(k); k) exceeds k, where (n; k) is the binomial coefficient (Ecklund et al. 1974, Erdős et al. 1993). The best lower bound known is

g(k)>=exp(csqrt((ln^3k)/(lnlnk)))

(Granville and Ramare 1996). Scheidler and Williams (1992) tabulated g(k) up to k=140, and Lukes et al. (1997) tabulated g(k) for 135<=k<=200. The values for n=1, 2, 3, ... are 3, 6, 7, 7, 23, 62, 143, 44, 159, 46, 47, 174, 2239, ... (OEIS A003458).

See also

Binomial Coefficient, Good Binomial Coefficient, Least Prime Factor

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References

Ecklund, E. F. Jr.; Erdős, P.; and Selfridge, J. L. "A New Function Associated with the prime factors of (n; k)." Math. Comput. 28, 647-649, 1974.Erdős, P.; Lacampagne, C. B.; and Selfridge, J. L. "Estimates of the Least Prime Factor of a Binomial Coefficient." Math. Comput. 61, 215-224, 1993.Granville, A. and Ramare, O. "Explicit Bounds on Exponential Sums and the Scarcity of Squarefree Binomial Coefficients." Mathematika 43, 73-107, 1996.Lukes, R. F.; Scheidler, R.; and Williams, H. C. "Further Tabulation of the Erdős-Selfridge Function." Math. Comput. 66, 1709-1717, 1997.Scheidler, R. and Williams, H. C. "A Method of Tabulating the Number-Theoretic Function g(k)." Math. Comput. 59, 251-257, 1992.Sloane, N. J. A. Sequence A003458/M2515 in "The On-Line Encyclopedia of Integer Sequences."

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Erdős-Selfridge Function

Cite this as:

Weisstein, Eric W. "Erdős-Selfridge Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Erdos-SelfridgeFunction.html

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