A327863 Numbers whose arithmetic derivative is a multiple of 3, cf. A003415.

0, 1, 8, 9, 14, 18, 20, 26, 27, 35, 36, 38, 44, 45, 50, 54, 62, 63, 64, 65, 68, 72, 74, 77, 81, 86, 90, 92, 95, 99, 108, 110, 112, 116, 117, 119, 122, 125, 126, 134, 135, 143, 144, 146, 153, 155, 158, 160, 161, 162, 164, 170, 171, 180, 185, 188, 189, 194, 196, 198, 203, 206, 207, 208, 209, 212, 215, 216, 218, 221, 225

Offset: 1

Author

Antti Karttunen, Sep 30 2019

Keywords

• nonn

Comments

From Antti Karttunen, May 27 2024 and Jun 12 2024: (Start)

This is a multiplicative semigroup: if m and n are in the sequence then so is m*n, and is generated by A008591 and A369659.

Term is present if and only if it is either a multiple of 9, or it is not a multiple of 3 and the sum of its prime factors (with repetition, A001414) is a multiple of 3, which happens iff the multiplicities of prime factors of the form 3m+1 (A002476) and of the form 3m-1 (A003627) are equal modulo 3.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001414, A002476, A003415, A003627, A235992, A289142, A327862, A327864, A327865, A359430 (characteristic function).

Positions of 0's in A373253.

Nonnegative integers are partitioned between this sequence, A373255, and A373257.

Disjoint union of A008591 and A369659.

Other subsequences: A327933, A369644, A370119, A373144, A373478, A373494, A373597.

Cf. also A369654, A370123.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A359430(n) = !(A003415(n)%3);
isA327863 = A359430;