OFFSET
1,2
COMMENTS
A multiplicative semigroup; if m and n are in the sequence then so is m*n.
Terms > 1 do not form a subsequence of A327934: Here 189 = 3^3 * 7 is present, although it is missing from A327934.
This is a subsequence of A046337, numbers with an even number of odd prime factors (with multiplicity). The semiprimes that occur here are all of the type (4m-1)*(4n+1), i.e., in A080774. A product of four odd primes (A046317) occurs here if either all of the primes have the same remainder modulo 4 (i.e., either all are of the type 4m-1 or all are of the type 4m+1), or two are of the other type, and two are of the other type. This follows because A003415(p*q*r*s) = (pqr + pqs + prs + qrs), while the product of four odd primes with just one prime of the different type are all located in A327862. - Antti Karttunen, Feb 05 2024
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..20000
EXAMPLE
189 = 3^3 * 7 has arithmetic derivative 189' = A003415(189) = 216 = 2^3 * 3^3. Because 189 is not a multiple of 4, but 216 is, 189 is included in this sequence.
MATHEMATICA
d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[600], ! Divisible[#, 4] && Divisible[d[#], 4] &] (* Amiram Eldar, Jan 31 2023 *)
PROG
(PARI) isA360110(n) = A360109(n);
CROSSREFS
Cf. A080774 (subsequence).
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 31 2023
STATUS
approved