cp's OEIS Frontend

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

%I A360110 #29 Feb 05 2024 18:09:24
%S A360110 1,15,35,39,51,55,81,87,91,95,111,115,119,123,143,155,159,183,187,189,
%T A360110 203,215,219,225,235,247,259,267,287,291,295,297,299,303,319,323,327,
%U A360110 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
%N A360110 Nonmultiples of 4 whose arithmetic derivative is a multiple of 4.
%C A360110 A multiplicative semigroup; if m and n are in the sequence then so is m*n.
%C A360110 Terms > 1 do not form a subsequence of A327934: Here 189 = 3^3 * 7 is present, although it is missing from A327934.
%C A360110 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
%H A360110 Antti Karttunen, <a href="/A360110/b360110.txt">Table of n, a(n) for n = 1..20000</a>
%e A360110 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.
%t A360110 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 *)
%o A360110 (PARI) isA360110(n) = A360109(n);
%Y A360110 Intersection of A327864 with A042968, or equally, with A046337.
%Y A360110 Setwise difference A046337 \ A327862.
%Y A360110 After 1, a subsequence of A327929 and of A099309.
%Y A360110 Subsequence of A235992, but not of A359829.
%Y A360110 Cf. A003415, A046317, A327934, A360109 (characteristic function).
%Y A360110 Cf. A080774 (subsequence).
%K A360110 nonn
%O A360110 1,2
%A A360110 _Antti Karttunen_, Jan 31 2023