A360110 Nonmultiples of 4 whose arithmetic derivative is a multiple of 4.

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

Offset: 1

Author

Antti Karttunen, Jan 31 2023

Keywords

• nonn

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

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Intersection of A327864 with A042968, or equally, with A046337.

Setwise difference A046337 \ A327862.

After 1, a subsequence of A327929 and of A099309.

Subsequence of A235992, but not of A359829.

Cf. A003415, A046317, A327934, A360109 (characteristic function).

Cf. A080774 (subsequence).

Programs

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 *)

PARI

isA360110(n) = A360109(n);