Tian Vlasic - cp's OEIS frontend

A369713 a(n) is the sum over all multiplicative partitions k of n of the absolute value of the Möbius function evaluated at k,n in the poset of multiplicative partitions of n under refinement.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 8, 2, 2, 2, 4, 1, 6, 1, 6, 2, 2, 2, 11, 1, 2, 2, 8, 1, 6, 1, 4, 4, 2, 1, 16, 2, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 16, 1, 2, 4, 11, 2, 6, 1, 4, 2, 6, 1, 24, 1, 2, 4, 4, 2, 6, 1, 16, 5, 2, 1, 16, 2

Offset: 1

Tian Vlasic, Jan 29 2024

nonn

If x and y are factorizations of the same integer and it is possible to produce x by further factoring the factors of y, flattening, and sorting, then x <= y.
For every natural number n, a(n) only depends on the prime signature of n.
a(n) is even if and only if n is a composite number.
Conjecture: There exists c such that a(n) <= n^c for all natural numbers n.

			The factorizations of 60 followed by their Moebius values are the following:
 (2*2*3*5) -> -3
 (2*2*15) ->  1
 (2*3*10) ->  2
 (2*5*6) ->  2
 (2*30) -> -1
 (3*4*5) ->  2
 (3*20) -> -1
 (4*15) -> -1
 (5*12) -> -1
 (6*10) -> -1
 (60) ->  1
Thus a(60)=16.

Cf. A001055, A002033, A025487, A045778, A050322, A064554, A077565, A097296, A190938, A216599, A317146.

A364656 Number of strict interval closure operators on a set of n elements.

Original entry on oeis.org

1, 1, 4, 45, 2062, 589602, 1553173541

Offset: 0

Tian Vlasic, Jul 31 2023

A closure operator cl on a set X is strict if the empty set is closed; it is an interval if for every subset S of X, the statement that for all x,y in S, cl({x,y}) is a subset of S implies that S is closed.

a(n) is also the number of interval convexities on a set of n elements (see Chepoi).

			The a(3) = 45 set-systems are the following ({} and {1,2,3} not shown).
    {1}   {1}{2}   {1}{2}{3}    {1}{2}{3}{12}   {1}{2}{3}{12}{13}
    {2}   {1}{3}   {1}{2}{12}   {1}{2}{3}{13}   {1}{2}{3}{12}{23}
    {3}   {2}{3}   {1}{2}{13}   {1}{2}{3}{23}   {1}{2}{3}{13}{23}
    {12}  {1}{12}  {1}{2}{23}   {1}{2}{12}{13}
    {13}  {1}{13}  {1}{3}{12}   {1}{2}{12}{23}
    {23}  {1}{23}  {1}{3}{13}   {1}{3}{12}{13}        {1}{2}{3}{12}{13}{23}
          {2}{12}  {1}{3}{23}   {1}{3}{13}{23}
          {2}{13}  {2}{3}{12}   {2}{3}{12}{23}
          {2}{23}  {2}{3}{13}   {2}{3}{13}{23}
          {3}{12}  {2}{3}{23}
          {3}{13}  {1}{12}{13}
          {3}{23}  {2}{12}{23}
                   {3}{13}{23}

G. M. Bergman. Lattices, Closure Operators, and Galois Connections. Springer, Cham. 2015. 173-212.

Victor Chepoi, Separation of Two Convex Sets in Convexity Structures
Dmitry I. Ignatov, Supporting iPython code for counting strict interval closure operators up to n=6, Github repository
Wikipedia, Closure operator

Cf. A334255, A358144, A358152, A356544.

Table[With[{closure = {X, set} |->
      Intersection @@ Select[X, SubsetQ[#, set] &]},
   Select[
    Select[
     Join[{{}, Range@n}, #] & /@ Subsets@Subsets[Range@n, {1, n - 1}],
      SubsetQ[#, Intersection @@@ Subsets[#, {2}]] &],
    X |->
     AllTrue[Complement[Subsets@Range@n, X],
      S |-> \[Not]
        AllTrue[Subsets[S, {1, 2}], SubsetQ[S, closure[X, #]] &]]]] //
   Length, {n, 4}]

New offset and a(5)-a(6) from Dmitry I. Ignatov, Nov 14 2023

A363876 Decimal expansion of the geometric mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

Original entry on oeis.org

9, 1, 6, 8, 1, 6, 9, 2, 3, 3, 8, 2, 1, 6, 8, 2, 4, 8, 1, 7, 5, 4, 6, 2, 5, 3, 8, 5, 7, 2, 3, 7, 0, 4, 0, 4, 5, 6, 7, 3, 5, 3, 2, 9, 4, 9, 9, 3, 7, 3, 6, 2, 4, 4, 3, 3, 7, 8, 4, 0, 1, 6, 6, 5, 1, 9, 8, 9, 0, 1, 3, 8, 4, 8, 1, 5, 9, 1, 0, 1, 0, 3, 4, 9, 0, 0, 0, 4

Offset: 0

Tian Vlasic, Jun 25 2023

The isoperimetric quotient of a curve is defined as Q = (4*Pi*A)/p^2, where A and p are the area and the perimeter of that curve respectively.

The isoperimetric quotient of an ellipse depends only on its eccentricity e in accordance to the formula Q = (Pi^2*sqrt(1-e^2))/(4*E(e)^2), where E() is the complete elliptic integral of the second kind.

			0.916816923382168248...

Eric Weisstein's World of Mathematics, Isoperimetric Quotient
Wikipedia, Elliptic integral

Cf. A102753, A363848, A363874.

First[RealDigits[Pi^2/2*Exp[-1 - 2*NIntegrate[Log[EllipticE[x^2]], {x, 0, 1}, WorkingPrecision -> 100]]]]

Equals ((Pi^2)/2) * exp(-1-2*Integral_{x=0..1} log(E(x)) dx).

A363874 Decimal expansion of the harmonic mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

Original entry on oeis.org

8, 7, 8, 9, 2, 0, 6, 5, 0, 8, 2, 9, 6, 0, 4, 1, 2, 4, 6, 2, 0, 2, 9, 7, 3, 2, 0, 0, 5, 3, 0, 7, 8, 4, 1, 6, 0, 2, 4, 9, 3, 3, 6, 4, 8, 6, 4, 2, 2, 9, 7, 7, 8, 0, 2, 0, 8, 9, 5, 7, 7, 3, 5, 2, 7, 1, 5, 0, 7, 2, 5, 3, 7, 1, 5, 9, 8, 8, 1, 9, 1, 8, 1, 8, 2, 8, 4, 3, 6

Offset: 0

Tian Vlasic, Jun 25 2023

The isoperimetric quotient of a curve is defined as Q = (4*Pi*A)/p^2, where A and p are the area and the perimeter of that curve respectively.

The isoperimetric quotient of an ellipse depends only on its eccentricity e in accordance to the formula Q = (Pi^2*sqrt(1-e^2))/(4*E(e)^2), where E() is the complete elliptic integral of the second kind.

			0.87892065082960412...

Eric Weisstein's World of Mathematics, Isoperimetric Quotient
Wikipedia, Elliptic integral

Cf. A091476, A363848, A363876.

First[RealDigits[Pi^2/(4 * NIntegrate[EllipticE[x^2]^2/Sqrt[1 - x^2], {x, 0, 1}, WorkingPrecision -> 100])]]

Pi^2/(4*intnum(x=0,1,(ellE(x)^2)/sqrt(1 - x^2))) \\ Hugo Pfoertner, Jun 25 2023

Equals Pi^2/(4*Integral_{x=0..1} (E(x)^2)/sqrt(1 - x^2) dx).

A363848 Decimal expansion of the arithmetic mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

Original entry on oeis.org

9, 3, 3, 1, 7, 4, 6, 5, 3, 4, 9, 8, 4, 6, 2, 6, 4, 4, 0, 1, 5, 5, 4, 4, 5, 3, 5, 2, 4, 8, 4, 6, 1, 0, 6, 1, 0, 8, 6, 7, 7, 3, 8, 5, 6, 2, 0, 1, 9, 3, 4, 9, 4, 3, 5, 9, 0, 1, 0, 3, 7, 9, 9, 8, 2, 3, 6, 3, 0, 9, 4, 1, 8, 6, 5, 4, 2, 6, 2, 0, 3, 4, 4, 7, 5, 1, 9, 6

Offset: 0

Tian Vlasic, Jun 24 2023

The isoperimetric quotient of a curve is defined as Q = (4*Pi*A)/p^2, where A and p are the area and the perimeter of that curve respectively.

The isoperimetric quotient of an ellipse depends only on its eccentricity e in accordance to the formula Q = (Pi^2*sqrt(1-e^2))/(4*E(e)^2), where E() is the complete elliptic integral of the second kind.

			0.933174653498462644...

Eric Weisstein's World of Mathematics, Isoperimetric Quotient
Wikipedia, Elliptic integral

Cf. A091476, A363874, A363876.

First[RealDigits[Pi^2/4 * NIntegrate[Sqrt[1-x^2]/EllipticE[x^2]^2, {x,0,1}, WorkingPrecision -> 100]]] (* Stefano Spezia, Jun 24 2023 *)

Equals ((Pi^2)/4) * Integral_{x=0..1} sqrt(1 - x^2)/E(x)^2 dx.

More terms from Stefano Spezia, Jun 24 2023

A358152 Number of strict closure operators on a set of n elements such that every point and every closed set not containing that point can be separated by clopen sets.

Original entry on oeis.org

1, 1, 2, 8, 121, 18460, 159273237

Offset: 0

Tian Vlasic, Nov 01 2022

A closure operator is strict if the empty set is closed.
A point p in X and a subset A of X not containing p are separated by a set H if p is an element of H and A is a subset of X\H.
Also the number of S_3 convexities on a set of n elements in the sense of Chepoi.

			The a(3) = 8 set-systems of closed sets:
  {{}, {1, 2, 3}}
  {{}, {1}, {2, 3}, {1, 2, 3}}
  {{}, {2}, {1, 3},{1, 2, 3}}
  {{}, {3}, {1, 2}, {1, 2, 3}}
  {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {1, 2, 3}}
  {{}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}}
  {{}, {1}, {2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
  {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

G. M. Bergman, "Lattices, Closure Operators, and Galois Connections", pp. 173-212 in "An Invitation to General Algebra and Universal Constructions", Springer, (2015).

Victor Chepoi, Separation of Two Convex Sets in Convexity Structures

Cf. A334255, A358144, A356544.

SeparatedPairQ[F_, n_] := AllTrue[
  Flatten[(x |-> ({x, #} & /@ Select[F, FreeQ[#, x] &])) /@ Range[n],
  1], MemberQ[F,
  _?(H |-> With[{H1 = Complement[Range[n], H]},
      MemberQ[F, H1] && MemberQ[H, #[[1]]
] && SubsetQ[H1, #[[2]]
]])]&];
Table[Length@Select[Select[
   Subsets[Subsets[Range[n]]],
   And[
     MemberQ[#, {}],
     MemberQ[#, Range[n]],
     SubsetQ[#, Intersection @@@ Tuples[#, 2]]] &
   ], SeparatedPairQ[#, n] &], {n, 0, 4}]

a(5)-a(6) from Christian Sievers, Jul 20 2024

A358144 Number of strict closure operators on a set of n elements such that all pairs of distinct points can be separated by clopen sets.

Original entry on oeis.org

1, 1, 1, 4, 167, 165791, 19194240969

Offset: 0

Tian Vlasic, Oct 31 2022

A closure operator is strict if the empty set is closed.
Two distinct points x,y in X are separated by a set H if x is an element of H and y is not an element of H.
Also the number of S_2 convexities on a set of n elements in the sense of Chepoi.

			The a(3) = 4 set-systems of closed sets:
  {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {1, 2, 3}}
  {{}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}}
  {{}, {1}, {2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
  {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

G. M. Bergman. Lattices, Closure Operators, and Galois Connections. Springer, Cham. 2015. 173-212 in "An Invitation to General Algebra and Universal Constructions", Springer, (2015).

Victor Chepoi, Separation of Two Convex Sets in Convexity Structures
Wikipedia, Closure operator

Cf. A334255, A358152, A356544.

SeparatedPairQ[F_, n_] := AllTrue[
Subsets[Range[n], {2}],
MemberQ[F,
_?(H |-> With[{H1 = Complement[Range[n], H]},
      MemberQ[F, H1] && MemberQ[H, #[[1]]
] && MemberQ[H1, #[[2]]
]])] &];
Table[Length@Select[Select[
   Subsets[Subsets[Range[n]]],
   And[
     MemberQ[#, {}],
     MemberQ[#, Range[n]],
     SubsetQ[#, Intersection @@@ Tuples[#, 2]]] &
   ], SeparatedPairQ[#, n] &] , {n, 0, 4}]

a(5) from Christian Sievers, Feb 04 2024

a(6) from Christian Sievers, Jun 13 2024

A356544 Number of strict closure operators on a set of n elements such that all pairs of nonempty disjoint closed sets can be separated by clopen sets.

Original entry on oeis.org

0, 1, 4, 35, 857, 84230, 70711467

Offset: 0

Tian Vlasic, Aug 11 2022

A closure operator is strict if the empty set is closed.
Two nonempty disjoint subsets A and B of X are separated by a set H if A is a subset of H and B is a subset of X\H.
Also the number of S_4 (Kakutani separation property) convexities on a set of n elements in the sense of Chepoi.

			The a(3) = 35 set-systems of closed sets:
{{}, {1, 2, 3}}
{{}, {1}, {1, 2, 3}}
{{}, {2}, {1, 2, 3}}
{{}, {3}, {1, 2, 3}}
{{}, {1, 2}, {1, 2, 3}}
{{}, {1, 3}, {1, 2, 3}}
{{}, {2, 3}, {1, 2, 3}}
{{}, {1}, {1, 2}, {1, 2, 3}}
{{}, {1}, {1, 3}, {1, 2, 3}}
{{}, {1}, {2, 3}, {1, 2, 3}}
{{}, {2}, {1, 2}, {1, 2, 3}}
{{}, {2}, {1, 3}, {1, 2, 3}}
{{}, {2}, {2, 3}, {1, 2, 3}}
{{}, {3}, {1, 2}, {1, 2, 3}}
{{}, {3}, {1, 3}, {1, 2, 3}}
{{}, {3}, {2, 3}, {1, 2, 3}}
{{}, {1}, {2}, {1, 3}, {1, 2, 3}}
{{}, {1}, {2}, {2, 3}, {1, 2, 3}}
{{}, {1}, {3}, {1, 2}, {1, 2, 3}}
{{}, {1}, {3}, {2, 3}, {1, 2, 3}}
{{}, {1}, {1, 2}, {1, 3}, {1, 2, 3}}
{{}, {2}, {3}, {1, 2}, {1, 2, 3}}
{{}, {2}, {3}, {1, 3}, {1, 2, 3}}
{{}, {2}, {1, 2}, {2, 3}, {1, 2, 3}}
{{}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
{{}, {1}, {2}, {1, 2}, {1, 3}, {1, 2, 3}}
{{}, {1}, {2}, {1, 2}, {2, 3}, {1, 2, 3}}
{{}, {1}, {3}, {1, 2}, {1, 3}, {1, 2, 3}}
{{}, {1}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
{{}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}}
{{}, {2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
{{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {1, 2, 3}}
{{}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}}
{{}, {1}, {2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
{{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

G. M. Bergman. Lattices, Closure Operators, and Galois Connections. Springer, Cham. 2015. 173-212.

Victor Chepoi, Separation of Two Convex Sets in Convexity Structures
Wikipedia, Closure operator

Cf. A334255, A358144, A358152.

SeparatedPairQ[A_][B_] := AnyTrue[A, And @@ MapThread[SubsetQ, {#, B}] &];
Table[Length[With[{X = Range[n]},
Select[Cases[Subsets@Subsets@X, {{}, _, X}],
   F |-> SubsetQ[F, Intersection @@@ Subsets[F, {2}]]
&& AllTrue[Select[Subsets[Drop[F, 1], {2}], Apply[DisjointQ]], SeparatedPairQ[Select[{#, Complement[X, #]} & /@ F, MemberQ[F, #[[2]]] &]]]]]], {n, 0, 4}]

a(5)-a(6) from Christian Sievers, Jun 13 2024

A350486 Numbers that have an equal number of even- and odd-length unordered factorizations and also an equal number of even- and odd-length unordered factorizations into distinct factors.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 180, 183, 185, 187, 192, 194

Offset: 1

Tian Vlasic, Jan 01 2022

First differs from A006881 at a(53) = 180.
By length, we mean the number of factors in a particular factorization.
Intersection of A319240 (factors are not necessarily distinct) and A319238 (factors are distinct).
Numbers k such that A316441(k) = A114592(k) = 0.
There are infinitely many terms in this sequence since all squarefree semiprimes (listed in A006881) are always such numbers.
There are no terms of the form p^k with p prime (listed in A000961).
Out of all numbers of the form p*q^k, p and q prime, only the numbers of the form p*q (A006881) and p*q^6 (A189987) are terms.
Similar methods can be applied to all prime signatures.

			6=2*3 (unrestricted) has an equal number (1) of even-length factorizations and odd-length factorizations, and 6=2*3 (distinct) has an equal number (1) of even-length factorizations and odd-length factorizations.

Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 2
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem

Cf. A006881, A114592, A189987, A316441, A319240, A319238.

facs[n_] := If[n <= 1, {{}}, Join @@ Table[Map[Prepend[#, d] &, Select[facs[n/d], Min @@ # >= d &]], {d, Rest[Divisors[n]]}]]; Intersection @@ First@Flatten[Position[#, 0] & /@ Transpose@Table[Sum[(-1)^Length[f], {f, #}] & /@ {facs[n], Select[facs[n], UnsameQ @@ # &]}, {n, #1, #2}], {3}]&[1,194] (* Robert P. P. McKone, Jan 05 2022, from Gus Wiseman in A319238 and A319240 *)

A349931 Numbers that have an equal number of factorizations of even and odd length in both unordered and ordered manners.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 63, 68, 72, 75, 76, 80, 92, 98, 99, 108, 112, 116, 117, 121, 124, 147, 148, 153, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 200, 207, 208, 212, 236, 240, 242, 244, 245, 252, 261, 268, 272, 275, 279, 284, 289, 292, 300

Offset: 1

Tian Vlasic, Dec 05 2021

Intersection of A319240 and A013929, i.e., terms of A319240 that are not squarefree.
A319240 lists the numbers that have an equal number of factorizations of even and odd length in an unordered manner.
A013929 lists the numbers that have an equal number of factorizations of even and odd length in an ordered manner.
There are infinitely many terms in this sequence since p^2 is always such a number for prime p.
Out of all numbers of the form p^k with p prime (listed in A000961), only the numbers of the form p^2 (A001248) are terms.
Out of all numbers of the form p*q^k, p and q prime, only the numbers of the form p*q (A006881), p*q^2 (A054753), p*q^4 (A178739) and p*q^6 (A189987) are terms.
Similar methods can be applied to all prime signatures.
Wilf's conjecture is equivalent to the statement that this sequence is the set difference of A319240 and A006881.

			12 = 2*6 = 3*4 = 2*2*3 (unordered) has an equal number (2) of even-length factorizations and odd-length factorizations, and 12 = 2*6 = 6*2 = 3*4 = 4*3 = 2*2*3 = 2*3*2 = 3*2*2 (ordered) has an equal number (4) of even-length factorizations and odd-length factorizations.

Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005.

Valerio De Angelis and Dominic Marcello, Wilf's Conjecture, The American Mathematical Monthly 123.6 (2016): 557-573.
S. de Wannemacker, T. Laffey and R. Osburn, On a conjecture of Wilf, arXiv:math/0608085 [math.NT], 2006-2007.

Cf. A006881, A013929, A074206, A319240, A316441, A000587.

f(n, m=n, k=0) = if(1==n, (-1)^k, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += f(n/d, d, k+1))); (s)); \\ A316441
isok(m) = (f(m) == 0) && ! issquarefree(m); \\ Michel Marcus, Dec 09 2021

cp's OEIS Frontend

User: Tian Vlasic

A369713 a(n) is the sum over all multiplicative partitions k of n of the absolute value of the Möbius function evaluated at k,n in the poset of multiplicative partitions of n under refinement.

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A364656 Number of strict interval closure operators on a set of n elements.

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A363876 Decimal expansion of the geometric mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

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A363874 Decimal expansion of the harmonic mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

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A363848 Decimal expansion of the arithmetic mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

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A358152 Number of strict closure operators on a set of n elements such that every point and every closed set not containing that point can be separated by clopen sets.

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Extensions

A358144 Number of strict closure operators on a set of n elements such that all pairs of distinct points can be separated by clopen sets.

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Mathematica

Extensions