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A273235
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Number of Ramanujan's largely composite numbers having prime(n) as the greatest prime divisor.
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0
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3, 10, 17, 28, 27, 43, 44, 69, 68, 58, 97, 97, 125, 164, 201, 185, 162, 254, 263, 313, 491, 434, 466, 417, 309, 358, 510, 633, 935, 1148, 454
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OFFSET
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1,1
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COMMENTS
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Theorem. The sequence is unbounded.
Proof. Since the sequence of highly composite numbers (A002182) is a subsequence of this sequence, it is sufficient to prove that the number M_n of highly composite numbers with the maximal prime divisor p_n is unbounded. Let N be a large highly composite number. Then for the greatest prime divisor p_N of N we have [Erdos] p_N=O(log N). So for all N<=x, p_N=O(log x).
If M_n=O(1), then the number of all highly composite numbers <=x is O(p_n)=O(log x). However, Erdos [Erdos] proved that this number is more than (log x)^(1+c) for a certain c>0.
So we have a contradiction. This means that M_n and this sequence are unbound. QED
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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