The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A003415

(Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

A003415

a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(m*n) = m*a(n) + n*a(m).
(history; published version)

#392 by Sean A. Irvine at Thu Aug 06 16:43:03 EDT 2026

STATUS

editing

approved

#391 by Sean A. Irvine at Thu Aug 06 16:41:39 EDT 2026

FORMULA

More generally, for any n > 1, a(n) >= W * n^(1-1/W), where W = A001222(n), with the equality attained at prime powers. Since the right-hand side is an increasing function of W, for a non-prime nonprime n, W >= 2 and thus a(n) >= 2*sqrt(n); for a non-prime nonprime and non-semiprime n, W >= 3 and thus a(n) >= 3*n^(2/3), etc. - Max Alekseyev, Sep 07 2025

STATUS

proposed

editing

Discussion

Thu Aug 06

16:43

Sean A. Irvine: Let's allow this one.

#390 by Michael S. Branicky at Thu Aug 06 16:05:20 EDT 2026

STATUS

editing

proposed

#389 by Michael S. Branicky at Thu Aug 06 12:22:13 EDT 2026

STATUS

proposed

editing

Discussion

Thu Aug 06

16:05

Michael S. Branicky: There is also an editorial push to keep core sequences to a manageable size, so let's see what other editors think.

#388 by Robert C. Lyons at Thu Aug 06 12:10:16 EDT 2026

STATUS

editing

proposed

Discussion

Thu Aug 06

12:22

Michael S. Branicky: If n = Product p_i^e_i, a(n) = n * Sum (e_i/p_i).
n = p^e ==> p^e * e / p = e* p^(e-1)

12:28

Darío Clavijo: Yes is a special case of the first formula: If n = Product p_i^e_i, a(n) = n * Sum (e_i/p_i) but it has a sum and a quotient a my formula expreses the special case in terms of powers and multiplications, that is why I think it has merits for its place. Since prime powers are a common special case, I thought this explicit formula might be useful to readers. But what editors deem appropiate is fine with me.

#387 by Robert C. Lyons at Thu Aug 06 12:10:00 EDT 2026

MAPLE

# Alternative:

MATHEMATICA

(* Alternative: *)

STATUS

proposed

editing

Discussion

Thu Aug 06

12:13

Michael S. Branicky: This is just a special case of the first formula, so not needed.

#386 by Darío Clavijo at Thu Aug 06 12:09:50 EDT 2026

STATUS

editing

proposed

Discussion

Thu Aug 06

12:10

Darío Clavijo: Found it. Thanks

#385 by Darío Clavijo at Thu Aug 06 12:09:46 EDT 2026

FORMULA

a(p^e) = e*(p^(e-1)) for p prime and e >= 1. - Darío Clavijo, Aug 6 06 2026

STATUS

proposed

editing

#384 by Darío Clavijo at Thu Aug 06 10:05:49 EDT 2026

STATUS

editing

proposed

Discussion

Thu Aug 06

10:22

Michel Marcus: Compare your signature with - _Franklin T. Adams-Watters_, Nov 07 2006

12:07

Darío Clavijo: Besides: f(Product_i p_i^e_i) = (Product_i p_i^e_i) * (Sum_i e_i*f(p_i)/p_i) which for me is not the same as a(p^e) = e*(p^(e-1)) for p prime and e >= 1 because it is a recurrence. I do not see the problem. I checked the concept and I see it ok and I checked the signature and see it ok. Maybe you can point me directly where the problem is. Thanks

12:08

Darío Clavijo: I also double checked the expresion a(p^e) = e*(p^(e-1)) for p prime and e >= 1 did not exist in any form.

#383 by Darío Clavijo at Thu Aug 06 10:05:36 EDT 2026

FORMULA

a(p^e) = e*(p^(e-1)) for p prime and e >= 1. - Darío Clavijo, Aug 6 2026

STATUS

approved

editing

Showing entries 1-10 | older changes