A331938 Indices of A002110(n) in A055932.

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

Offset: 0

Author

Michael De Vlieger, Feb 02 2020

Keywords

• nonn
• more

Comments

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.

Offset is 0 since A002110(0) = 1.

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.

Crossrefs

Cf. A001221, A002110, A055932, A067255.

Programs

Mathematica

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] ]

Extensions

a(21)-a(26) from Giovanni Resta, Feb 03 2020