|
|
A360110
|
|
Nonmultiples of 4 whose arithmetic derivative is a multiple of 4.
|
|
10
|
|
|
1, 15, 35, 39, 51, 55, 81, 87, 91, 95, 111, 115, 119, 123, 143, 155, 159, 183, 187, 189, 203, 215, 219, 225, 235, 247, 259, 267, 287, 291, 295, 297, 299, 303, 319, 323, 327, 335, 339, 355, 371, 391, 395, 403, 407, 411, 415, 427, 441, 447, 451, 471, 511, 513, 515, 519, 525, 527, 535, 543, 551, 559, 579
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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
|
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|