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A108795
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Conjectured greatest number k such that C(2k,k) is not divisible by any odd prime to the n-th power.
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0
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OFFSET
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1,2
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COMMENTS
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Checked by Jack Brennen to 5.93*10^10 and in fact every number beyond 14384056005 was divisible by at least two odd-prime-fourth-powers. C(2*14384056005,14384056005) seems to be the last such number which is only divisible by a single odd-prime-fourth-power, being divisible by 5^9 but by no other prime more than 3 times.
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, C33.
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LINKS
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Table of n, a(n) for n=1..4.
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EXAMPLE
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a(1)=1 because for all k's>1 C(2k,k) is divisible by an odd prime.
a(2)=786 because it is the last entry in A059097, i.e., C(1572,786) has no prime factor squared.
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MATHEMATICA
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expoPF[k_, n_] := Module[{s = 0, x = n}, While[x > 0, x = Floor[x/k]; s += x]; s]; goodQ[n_] := Module[{i = 2, p}, While[p = Prime[i]; p <= n && expoPF[p, 2n] < 3 + 2expoPF[p, n], i++ ]; p > n]; Do[ If[ goodQ[n], Print[n]], {n, 5500000}]
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CROSSREFS
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Cf. A059097.
Sequence in context: A031734 A097776 A031526 * A097774 A031896 A045231
Adjacent sequences: A108792 A108793 A108794 * A108796 A108797 A108798
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KEYWORD
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nonn
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AUTHOR
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R. K. Guy and Robert G. Wilson v, Nov 29 2005
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EXTENSIONS
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a(4) from Jack Brennen, Nov 30 2005
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STATUS
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approved
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