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A327863
Numbers whose arithmetic derivative is a multiple of 3, cf. A003415.
19
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
1,3
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
PROG
(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;
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.
Sequence in context: A096784 A175859 A059869 * A213016 A373478 A105833
KEYWORD
nonn
AUTHOR
Antti Karttunen, Sep 30 2019
STATUS
approved