A328520 GCD of terms in A002182 that have n prime factors counted with multiplicity.

1, 2, 2, 12, 12, 12, 60, 120, 2520, 2520, 2520, 55440, 55440, 720720, 720720, 12252240, 36756720, 698377680, 3491888400, 80313433200, 160626866400, 160626866400, 9316358251200, 288807105787200, 2021649740510400, 74801040398884800, 74801040398884800, 3066842656354276800, 131874234223233902400

Offset: 0

Author

David A. Corneth, Jan 04 2020

Keywords

• nonn

Comments

If for every term t > 1 in A002182 there exists a term in A002182 of the form t/p for some prime p|t (Cf. A328523) then (1): a(n+1) is a multiple of a(n) for n >= 1 and (2): this sequence can be used to find terms to A199337 and A330737.

Proof of (1): Let Generation(n) be the terms in A002182 with n prime factors counted with multiplicity. Then a(n) = GCD(Generation(n)). As each term in Generation(n) is of the form a(n) * t for some t and each term in Generation(n + 1) is p * g for some g in Generation(n), a(n + 1) is a multiple of a(n).

Proof of (2): As a(n + 1) is a multiple of a(n) we have that if m | a(n), we also have m | a(n + k), k >= 0 hence the largest number not divisible by m can have at most n - 1 prime factors counted with multiplicity.

Given Generation(n) we can find all candidates for Generation(n + 1) and from there on find terms to A002182, A199337 and A330737.

Examples

			The terms in A002182 with n = 4 prime divisors counted with multiplicity are 24, 36 and 60. Their GCD is 12 hence a(4) = 12.
Furthermore, If for every term t > 1 in A002182 there exists a term in A002182 of the form t/p for some prime p|t then we have that each term with more than 4 prime divisors counted with multiplicity is a multiple of at least one of 24, 36 or 60 hence is divisible by 12.
		

Crossrefs

Cf. A001222, A002182, A199337, A328523, A330737

Cf. also A054481.

Programs

Mathematica

(* First, load the function f at A025487, then: *)
Block[{s = Union@ Flatten@ f@ 20, t}, t = DivisorSigma[0, s]; s = Map[s[[FirstPosition[t, #][[1]] ]] &, Union@ FoldList[Max, t]]; t = PrimeOmega[s]; Drop[Array[GCD @@ s[[Position[t, #][[All, 1]] ]] &, Max@ t + 1, 0], -3] ] (* Michael De Vlieger, Jan 12 2020 *)