

A369659


Nonmultiples of 3 whose arithmetic derivative, or equally, the sum of prime factors (with multiplicity) is a multiple of 3.


8



1, 8, 14, 20, 26, 35, 38, 44, 50, 62, 64, 65, 68, 74, 77, 86, 92, 95, 110, 112, 116, 119, 122, 125, 134, 143, 146, 155, 158, 160, 161, 164, 170, 185, 188, 194, 196, 203, 206, 208, 209, 212, 215, 218, 221, 230, 236, 242, 254, 275, 278, 280, 284, 287, 290, 299, 302, 304, 305, 314, 323, 326, 329, 332, 335, 341, 343
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OFFSET

1,2


COMMENTS

This is a subsequence of A373475, containing all its terms that are not multiples of 3. (See comments in A373475 for a proof). The first difference from A373475 is at n=4186, where A373475(4186) = 19683 = 3^9, the value which is missing from this sequence.  Antti Karttunen, Jun 07 2024
A multiplicative semigroup: if m and n are in the sequence, then so is m*n.
Numbers that are not multiples of 3, and the multiplicities of prime factors of the forms 3m+1 (A002476) and 3m1 (A003627) are equal modulo 3.
Like A373597, which is a subsequence, also this sequence can be viewed as a kind of k=3 variant of A046337.
A289142, numbers whose sum of prime factors (with multiplicity, A001414) is a multiple of 3, is generated (as a multiplicative semigroup) by the union of this sequence with {3}.
A327863, numbers whose arithmetic derivative is a multiple of 3, is generated by this sequence and A008591.
A373478, numbers that are in the intersection of A289142 and A327863, is generated by the union of this sequence with {9, 27}.
A373475, numbers that are in the intersection of A289142 and A369644 (positions of multiples of 3 in A083345), is generated by the union of this sequence with {19683}, where 19683 = 3^9.
(End)
The integers in the multiplicative subgroup of positive rationals generated by semiprimes of the form 3m+2 (A344872) and cubes of primes except 27.  Peter Munn, Jun 19 2024


LINKS



EXAMPLE

280 = 2*2*2*5*7 is included as it is not a multiple of 3, and one of its prime factors (7) is of the form 3m+1 and four are of the form 3m1, and because 4 == 1 (mod 3). Also, A001414(280) = 18, and A003415(280) = 516, both of which are multiples of 3.  Antti Karttunen, Jun 12 2024


PROG



CROSSREFS

Cf. also A046337, A360110, A369969 for cases k=2, 4, 5 of "Nonmultiples of k whose arithmetic derivative is a multiple of k".


KEYWORD

nonn


AUTHOR



EXTENSIONS

Name amended with an alternative definition by Antti Karttunen, Jun 11 2024


STATUS

approved



