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Revision History for A081707 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

a(n) = tau(n) - bigomega(n) = A000005(n) - A001222(n).
(history; published version)

#36 by Joerg Arndt at Sat May 30 11:15:05 EDT 2020

FORMULA

If g(n) is the Dirichlet inverse of the sequence omega(n)+1 = A001221(n)+1, then this sequence is A008836(n)(Sum_{d|n} g(d)), i.e., it is the Liouville lambda function scaled by the Dirichlet convolution of g with itself. The inverse sequence starts as {g(n)}_{n >= 1} = {1, -2, -2, 2, -2, 5, -2, -2, 2, 5, -2, -7, -2, 5, 5, 2, -2, -7, -2, -7, 5, 5, -2, 9, ... }. For squarefree n, the function g(n) = A008836(n) A000522(n). The last observation can be proved, but is tedious. - Maxie D. Schmidt, Feb 21 2020

KEYWORD

nonn,changed

nonn

STATUS

proposed

approved

#35 by Joerg Arndt at Mon May 25 09:28:02 EDT 2020

STATUS

editing

proposed

Discussion

Mon May 25

10:31

Michel Marcus: in "For squarefree n, the function g(n) = A008836(n) A000522(n)" there is no symbol between A008836 and A000522 ??

Sat May 30

11:14

Joerg Arndt: I'll revert now.

#34 by Joerg Arndt at Mon May 25 09:26:45 EDT 2020

FORMULA

If g(n) is the Dirichlet inverse of the sequence omega(n)+1 = A001221(n)+1, then this sequence is A008836(n)(Sum_{d|n} g(d)), i.e., it is the Liouville lambda function scaled by the Dirichlet convolution of g with itself. The inverse sequence starts as {g(n)}_{n >= 1} = {1, -2, -2, 2, -2, 5, -2, -2, 2, 5, -2, -7, -2, 5, 5, 2, -2, -7, -2, -7, 5, 5, -2, 9, ... }. For squarefree n, the function g(n) = A008836(n) A000522(n). The last observation can be proved, but is tedious. _- _Maxie D. Schmidt_, Feb 21 2020

Discussion

Mon May 25

09:28

Joerg Arndt: Sitting forever...  Moving to reviewing queue.  To me the new comment looks dubious; leaving for the editor who know more of the topic at hand.

#33 by Maxie D. Schmidt at Fri Feb 21 00:34:36 EST 2020

FORMULA

If g(n) is the Dirichlet inverse of the sequence omega(n)+1 = A001221(n)+1, then this sequence is A008836(n)(Sum_{d|n} g(d)), i.e., it is the Liouville lambda function scaled by the Dirichlet convolution of g with itself. The inverse sequence starts as {g(n)}_{n >= 1} = {1, -2, -2, 2, -2, 5, -2, -2, 2, 5, -2, -7, -2, 5, 5, 2, -2, -7, -2, -7, 5, 5, -2, 9, ... }. For squarefree n, the function g(n) = A008836(n) A000522(n). The last observation can be proved, but is tedious. _Maxie D. Schmidt_, Feb 21 2020

STATUS

proposed

editing

Discussion

Fri May 22

12:26

OEIS Server: This sequence has not been edited or commented on for a week
yet is not proposed for review.  If it is ready for review, please
visit https://oeis.org/draft/A081707 and click the button that reads
"These changes are ready for review by an OEIS Editor."

Thanks.
  - The OEIS Server

#32 by Maxie D. Schmidt at Thu Feb 20 21:07:49 EST 2020

STATUS

editing

proposed

Discussion

Thu Feb 20

23:40

Andrew Howroyd: Most of your comment seems to pertain more to g(n) than this sequence. For example "For squarefree n, the function g(n) = A008836(n) A000522(n)" doesn't involve this sequence at all.

23:47

Andrew Howroyd: Also is "(Sum_{d|n} g(d))" the same as "the Dirichlet convolution of g with itself"? (shouldn't convolution with itself be Sum_{d|n} g(d)*g(n/d) ??)

Fri Feb 21

00:23

Michel Marcus: g(n) = A008836(n) A000522(n) you mean g(n) = A008836(n)*A000522(n) ?

00:24

Michel Marcus: A008836(n)(Sum_{d|n} g(d)) : you mean A008836(n)*(Sum_{d|n} g(d)) ??

00:25

Michel Marcus: The comment should be signed : This is done by appending " - ~~~~" at the end of the line

00:26

Michel Marcus: But then why don't you submit 1, -2, -2, 2, -2, 5, -2, -2, 2, 5, -2, -7, -2, 5, 5, 2, -2, -7, -2, -7, 5, 5, -2, 9, ... as a new sequence and add your comments there ?

00:32

Maxie D. Schmidt: @Andrew: A convolution of g with itself yields this sequence scaled by the Liouville lambda function. It's definitely related, and it turns out that one of the functions in the original sequence difference has a (shifted) Dirichlet inverse that generates the g(n). If I went hunting for this sequence in OEIS, I would want to see the elaboration on g(n)...

For squarefree n, this lets you express the original sequence (through Mobius inversion) in terms of a nicer sequence related to derangements / subfactorials. This is a useful property. Again, if I went hunting down this sequence, I'd be pleased to know this level of detail to other methods of generating the sequence at hand.

I misspoke: The convolution should be (g * 1)(n). So yes.

#31 by Maxie D. Schmidt at Thu Feb 20 21:06:23 EST 2020

FORMULA

If g(n) is the Dirichlet inverse of the sequence omega(n)+1 = A001221(n)+1, then this sequence is A008836(n)(Sum_{d|n} g(d)), i.e., it is the Liouville lambda function scaled by the Dirichlet convolution of g with itself. The inverse sequence starts as {g(n)}_{n >= 1} = {1, -2, -2, 2, -2, 5, -2, -2, 2, 5, -2, -7, -2, 5, 5, 2, -2, -7, -2, -7, 5, 5, -2, 9, ... }. For squarefree n, the function g(n) = A008836(n) A000522(n). The last observation can be proved, but is tedious.

STATUS

approved

editing

Discussion

Thu Feb 20

21:07

Maxie D. Schmidt: This sequence is actually very interesting and related to some work I'm doing on the Mertens function. Please accept these edits :)

#30 by Alois P. Heinz at Tue Nov 24 18:02:34 EST 2015

STATUS

proposed

approved

#29 by Geoffrey Critzer at Tue Nov 24 17:50:24 EST 2015

STATUS

editing

proposed

#28 by Geoffrey Critzer at Tue Nov 24 05:36:09 EST 2015

EXAMPLE

After first statement in comment section, a(60) = 8 because we have: 1,6,10,12,15,20,30,60. We have excluded the The divisors 2,3,4,5 are excluded from the count. - Geoffrey Critzer, Nov 22 2015

STATUS

proposed

editing

Discussion

Tue Nov 24

05:38

Geoffrey Critzer: @ Michel Marcus Yes, thanks.

#27 by Michel Marcus at Tue Nov 24 02:44:41 EST 2015

STATUS

editing

proposed