The On-Line Encyclopedia of Integer Sequences (OEIS)

Revisions by Benedict W. J. Irwin (See also Benedict W. J. Irwin's wiki page)

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

A143519

Moebius transform of A010051, the characteristic function of the primes: a(n) = Sum_{d|n} mu(n/d)*A010051(d); A054525 * A010051.
(history; published version)

#40 by Benedict W. J. Irwin at Wed Mar 02 05:28:40 EST 2022

STATUS

editing

proposed

Discussion

Wed Mar 02

09:19

Michel Marcus: use x,y,z rather than a,b,c ??

A345354

a(n) = Sum_{p|n, p prime} omega(n/p).
(history; published version)

#15 by Benedict W. J. Irwin at Wed Mar 02 05:17:16 EST 2022

STATUS

editing

proposed

Discussion

Wed Mar 02

09:19

Michel Marcus: use x,y,z rather than a,b,c ??

#14 by Benedict W. J. Irwin at Wed Mar 02 05:16:46 EST 2022

FORMULA

a(n) = Sum_{a*b*c=n} omega(a)*omega(b)*mu(c). - Benedict W. J. Irwin, Mar 02 2022

STATUS

approved

editing

A143519

Moebius transform of A010051, the characteristic function of the primes: a(n) = Sum_{d|n} mu(n/d)*A010051(d); A054525 * A010051.
(history; published version)

#39 by Benedict W. J. Irwin at Wed Mar 02 05:11:03 EST 2022

FORMULA

a(n) = Sum_{a*b*c=n} omega(a)*mu(b)*mu(c). - Benedict W. J. Irwin, Mar 02 2022

STATUS

approved

editing

A007428

Moebius transform applied thrice to sequence 1,0,0,0,....
(history; published version)

#51 by Benedict W. J. Irwin at Wed Mar 02 05:03:07 EST 2022

STATUS

editing

proposed

Discussion

Wed Mar 02

09:20

Michel Marcus: use x,y,z rather than a,b,c ??

#50 by Benedict W. J. Irwin at Wed Mar 02 05:02:37 EST 2022

FORMULA

a(n) = Sum_{a*b*c=n} mu(a)*mu(b)*mu(c). - Benedict W. J. Irwin, Mar 02 2022

STATUS

approved

editing

A301737

Denominator of cumulative weight of certain D-forests on n nodes.
(history; published version)

#23 by Benedict W. J. Irwin at Fri Jul 31 10:52:23 EDT 2020

COMMENTS

a(n) has a lot of overlap with coefficients of [x in expression Table[Sum[Pochhammer[] sum_{k=0}^n x,^(k]FactorialPower[) 1,_(n-k), with [x],{ the coefficient of x, x^(k) the rising factorial, and x_(k,0,n}],{n,0,12}] // Expand // Column) the falling factorial. Benedict W. J. Irwin, Jul 28 2020

Discussion

Sat Nov 21

11:28

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/A301737 and click the button that reads
"These changes are ready for review by an OEIS Editor."

Thanks.
  - The OEIS Server

A124385

For all n >= 2, Sum_{2<=k<=n, gcd(k,n)>1} a(k) = 1. a(1)=1.
(history; published version)

#13 by Benedict W. J. Irwin at Fri Jul 31 10:27:35 EDT 2020

COMMENTS

In analogy to the first column of the inverse of the lower triangular divisor matrix, A051731, giving the Moebius mu function, A008683, this sequence appears to match (up to sign) the first column of the inverse of the lower triangular matrix where entries are 1 if i and j share a common prime factor, A304569. Benedict W. J. Irwin, Jul 31 2020

STATUS

approved

editing

Discussion

Fri Jul 31

10:28

Benedict W. J. Irwin: F[a_,b_]:=If[a>b,If[b==1||Length[Intersection[FactorInteger[a][[All,1]]
,FactorInteger[b][[All,1]]]]>0,1,0],If[a==b,1,0]]

Inverse[Table[F[a,b],{a,1,100},{b,1,100}]][[All,1]]

Sat Nov 21

11:28

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/A124385 and click the button that reads
"These changes are ready for review by an OEIS Editor."

Thanks.
  - The OEIS Server

A301737

Denominator of cumulative weight of certain D-forests on n nodes.
(history; published version)

#22 by Benedict W. J. Irwin at Tue Jul 28 13:26:46 EDT 2020

COMMENTS

a(n) has a lot of overlap with coefficients of x in expression Table[Sum[Pochhammer[x,k]FactorialPower[1,n-k],{k,0,n}],{n,0,12}] // Expand // Column. Benedict W. J. Irwin, Jul 28 2020

STATUS

approved

editing

Discussion

Fri Jul 31

01:52

Joerg Arndt: More than one problem here: a(n) denoted the term with index n, turning the whole comment rather meaningless; Mathematica code is not OK as substitute for a formula.

10:45

Benedict W. J. Irwin: Hi Joerg, long time. Yes, I was not satisfied with this yet and was working to improve the expression. I wouldn't say it's 'meaningless' as it's a fractional sequence, so there could well be a cancellation with the numerator as often seen in exponential type generating functions. There is a combinatoric justification/interpretation as this denominator looks like a binomial style convolution of rising and lowering factorials, whereas in the exponential generating function case, it's a binomial convolution with unsigned Stirling numbers and 'unity'.

Best.

A037256

a(n) = n!*Sum_{i=0..n-1} (n-i)*(-2)^i/(i+1)!.
(history; published version)

#43 by Benedict W. J. Irwin at Mon Jul 06 09:17:21 EDT 2020

STATUS

editing

proposed