A369659 Non-multiples of 3 whose arithmetic derivative, or equally, the sum of prime factors (with multiplicity) is a multiple of 3.

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

Offset: 1

Author

Antti Karttunen, Feb 10 2024

Keywords

• nonn

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

From Antti Karttunen, Jun 11 2024: (Start)

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 3m-1 (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

Examples

			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 3m-1, 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
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..33911
• Index entries for sequences which agree for a long time but are different

Crossrefs

Cf. A001414, A002476, A003415, A003627, A083345, A369658 (characteristic function).

Intersection of A001651 and A327863.

Intersection of A001651 and A373475.

Setwise difference A373475 \ A373476.

Subsequence of A369644, which is a subsequence of A327863, and also of the following sequences: A289142, A373475, A373478.

Includes A030078 \ {27}, A344872 and A373597 as subsequences.

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

Cf. also A374044.

Programs

PARI

\\ See A369658.

Extensions

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