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1, 2, 4, 10, 28, 83, 227, 626, 1644, 4290, 11322, 28965, 74469, 189436, 471910, 1166247, 2884920, 7130085, 17349489, 42180190, 101820577, 242907065, 579402163, 1375056009, 3262651085, 7768448187, 18411720785
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OFFSET
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0,2
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COMMENTS
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A055932 lists numbers m whose prime divisors p are consecutive primes starting with 2, admitting multiplicity, while A002110 lists numbers m that are products of distinct consecutive primes starting with 2. Therefore, A002110 is a subset of A055932.
Let 0 <= i <= k, integers. We can write an efficient algorithm to construct a complete list of all terms m of A055932 up to A002110(k) using A067255(m). Every term m in the list has omega(m) = A001221(m) <= k. Starting with A002110(i), we use A067255 to encode m, i.e., the list of multiplicities e pertaining to the 1st..i-th prime p_i, allowing position of the multiplicity e in the list to convey p_i. Thus, the first "recipe" for m = A002110(i) = {1, 1, ..., 1}, a list of i ones. If this does not exceed the limit A002110(k), then we accept it as a value, then increment the last multiplicity. When we have an invalid recipe, we increment the penultimate multiplicity and reset the last to 1, etc., until we have generated all m <= A002110(k). As a measure of efficiency, this algorithm generates 1 <= m <= A002110(12), 74469 terms, in about 2 seconds including sorting, on a 64-bit Intel Xeon E-2286M (2.40 GHz) processor. This is the same amount of time it takes to test numbers 1..400000 to yield the 575 smallest terms of the same sequence.
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LINKS
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MATHEMATICA
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With[{s = Import["https://oeis.org/A002110/b002110.txt", "Data"][[1 ;; 8, -1]], t = TakeWhile[Import["https://oeis.org/A055932/b055932.txt", "Data"], Length@ # > 0 &][[All, -1]]}, TakeWhile[Map[FirstPosition[t, #][[1]] &, s], IntegerQ] ]
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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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EXTENSIONS
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STATUS
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approved
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