A327933 -id:A327933 - cp's OEIS frontend

A235992 Numbers with an even arithmetic derivative, cf. A003415.

0, 1, 4, 8, 9, 12, 15, 16, 20, 21, 24, 25, 28, 32, 33, 35, 36, 39, 40, 44, 48, 49, 51, 52, 55, 56, 57, 60, 64, 65, 68, 69, 72, 76, 77, 80, 81, 84, 85, 87, 88, 91, 92, 93, 95, 96, 100, 104, 108, 111, 112, 115, 116, 119, 120, 121, 123, 124, 128, 129, 132, 133

Offset: 1

Reinhard Zumkeller, Mar 11 2014

nonn

A165560(a(n)) = 0; A003415(a(n)) mod 2 = 0.
For n > 1: A007814(a(n)) <> 1, A006519(a(n)) <> 2.
Union of multiples of 4 and odd numbers with an even number of prime factors with multiplicity. - Charlie Neder, Feb 25 2019
After two initial terms (0 and 1), numbers n such that A086134(n) = 2. - Antti Karttunen, Sep 30 2019
A multiplicative semigroup; if m and n are in the sequence then so is m*n. (See also comments in A359780.) - Antti Karttunen, Jan 17 2023

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A235991 (complement).
Union of A327862 and A327864.
Union of A359829 (primitive elements) and A359831 (nonprimitive elements).
Cf. A003415, A086134, A327863, A327865, A327933, A327935, A358680 (characteristic function).
Positions of multiples of 4 in A358669 (and in A358765).
Cf. also A028260, A036349, A046337, A332820 (other multiplicative semigroups), and comments in A359780.

a235992 n = a235992_list !! (n-1)
a235992_list = filter (even . a003415) [0..]

Select[Range[0, 133], EvenQ@ If[Abs@ # < 2, 0, # Total[#2/#1 & @@@ FactorInteger[Abs@ #]]] &] (* Michael De Vlieger, Sep 30 2019 *)

from itertools import count, islice
from sympy import factorint
def A235992_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n: not n&3 or (n&1 and not sum(factorint(n).values())&1), count(max(startvalue,0)))
A235992_list = list(islice(A235992_gen(),40)) # Chai Wah Wu, Nov 04 2022

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

Original entry on oeis.org

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

Antti Karttunen, Sep 30 2019

nonn

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)

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

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.

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;

A327935 Numbers for which the smallest prime factor of their arithmetic derivative is 5.

Original entry on oeis.org

6, 46, 75, 106, 150, 166, 175, 226, 250, 266, 325, 346, 350, 406, 429, 466, 475, 526, 546, 550, 586, 646, 650, 682, 706, 750, 759, 766, 775, 826, 847, 850, 886, 925, 950, 966, 1006, 1050, 1075, 1083, 1106, 1126, 1150, 1186, 1209, 1246, 1250, 1254, 1306, 1326, 1342, 1366, 1406, 1419, 1421, 1450, 1486, 1525, 1526, 1546

Offset: 1

Antti Karttunen, Sep 30 2019

nonn

Numbers n for which A086134(n) = 5.

Numbers whose arithmetic derivative is an odd multiple of five, but is not a multiple of three.

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

Cf. A003415, A086134, A235992, A327933.

Subsequence of A235991, and also of A327865.

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A086134(n) = { my(d=A003415(n)); if(d<=1,0,factor(d)[1, 1]); };
isA327935(n) = (5==A086134(n));

from itertools import count, islice
from sympy import factorint
def A327935_gen(startvalue=2): # generator of terms  >= startvalue
    return filter(lambda n: (m:=sum((n*e//p for p,e in factorint(n).items())))&1 and m%3 and not m%5, count(max(startvalue,2)))
A327935_list = list(islice(A327935_gen(),40)) # Chai Wah Wu, Nov 04 2022

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A235992 Numbers with an even arithmetic derivative, cf. A003415.

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

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A327935 Numbers for which the smallest prime factor of their arithmetic derivative is 5.

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