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Revision History for A068311

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A068311

Arithmetic derivative of n!.
(history; published version)

#34 by Alois P. Heinz at Tue Feb 06 07:56:22 EST 2024

STATUS

editing

approved

#33 by Alois P. Heinz at Tue Feb 06 07:56:17 EST 2024

FORMULA

a(n) = A003415(A000142(n)).

CROSSREFS

Equals A003415(A000142(n)).

Cf. A000142, A003415.

STATUS

approved

editing

#32 by Alois P. Heinz at Fri Oct 27 19:37:59 EDT 2023

STATUS

editing

approved

#31 by Alois P. Heinz at Fri Oct 27 19:37:54 EDT 2023

LINKS

T. D. Noe and Alois P. Heinz, <a href="/A068311/b068311.txt">Table of n, a(n) for n = 0..450</a> (first 101 terms from T. D. Noe)

STATUS

approved

editing

#30 by Alois P. Heinz at Mon Jun 13 13:00:52 EDT 2022

STATUS

reviewed

approved

#29 by Michel Marcus at Mon Jun 13 12:51:06 EDT 2022

STATUS

proposed

reviewed

#28 by Chai Wah Wu at Mon Jun 13 10:14:39 EDT 2022

STATUS

editing

proposed

Discussion

Mon Jun 13

11:08

Michel Marcus: I am using https://sagecell.sagemath.org/  ...  (version is ???)

12:00

Michel Marcus: apparently it is 3.10 ??  /home/sc_serv/sage/local/var/lib/sage/venv-python3.10.3/lib/python3.10/lib-dynload/math.cpython-310-x86_64-linux-gnu.so

12:34

Chai Wah Wu: @Michel. Maybe this is a difference between Sage and Python? I was able to run this on the official online python shell at https://www.python.org/shell/
I think the code is reinterpreted by Sage to call Python. On sagecell I get the same error message as you and the error occurs when running the line:
--> 576         return sum(expression, *args)
so maybe Sage run the python sum function without passing it the keyword start. I don't use Sage, so I don't know the difference between Sage and Python.

12:51

Michel Marcus: ok then

#27 by Chai Wah Wu at Mon Jun 13 09:56:35 EDT 2022

PROG

(Python 3.8+)

STATUS

proposed

editing

Discussion

Mon Jun 13

10:14

Chai Wah Wu: @Michel are you using Python version 3.8 or up? The start parameter in sum was introduced in 3.8. I added that info into the header.

#26 by Jon E. Schoenfield at Sun Jun 12 21:50:01 EDT 2022

STATUS

editing

proposed

Discussion

Mon Jun 13

00:00

Michel Marcus: TypeError: unsupported operand type(s) for +: 'int' and 'Counter'

#25 by Jon E. Schoenfield at Sun Jun 12 21:49:59 EDT 2022

EXAMPLE

= (1*3 + 2*1)*4 + 6*(2*2*1) = 5*4 + 6*4 = 44;

where d(n) = A003415(n) with d(1)=0, d(prime)=1 and d(m*n)= d (m)*n + m*d(n).

a(6)=2064 because the arithmetic derivative of 6!=720 is 720*(4/2 + 2/3 + 1/5).

PROG

(MAGMAMagma) Ad:=func<h | h*(&+[Factorisation(h)[i][2]/Factorisation(h)[i][1]: i in [1..#Factorisation(h)]])>; [n le 1 select 0 else Ad(Factorial(n)): n in [0..30]]; // Bruno Berselli, Oct 23 2013

STATUS

proposed

editing