Elijah Beregovsky - cp's OEIS frontend

A384134 Triangle read by rows: T(n,k) is the number of Cauchy-complete categories with n morphisms and k objects.

1, 1, 1, 1, 2, 1, 2, 6, 2, 1, 1, 12, 9, 2, 1, 2, 23, 25, 10, 2, 1, 1, 45, 69, 35, 10, 2, 1, 5, 98, 178, 119, 38, 10, 2, 1, 2, 278, 457, 371, 151, 39, 10, 2, 1

Offset: 1

Elijah Beregovsky, May 20 2025

A Cauchy complete (also called Karoubi complete or idempotent-complete) category is one in which all idempotents split. In other words, in a Cauchy-complete category every arrow e:A->A such that e=e*e has a retract, meaning there exists an object B and morphisms r:A→B and s:B→A such that s∘r=e but r∘s=1_B.

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  2,  6,  2,  1;
  1, 12,  9,  2, 1;
  2, 23, 25, 10, 2, 1;
  ...

Geoff Cruttwell, Counting Finite Categories, presentation, (2018).
nLab, Cauchy complete category.
nLab, Karoubi envelope.

Cf. A384135 (row sums), A000001 (column 1), A384066 (limiting values), A125697.

T(n,k) = A384066(n-k) if k >= (2/3)*n.
T(3n,2n) = T(3n-1,2n-1) + 1 when n >= 1.
T(3n-1,2n-1) = T(3n-2,2n-2) + 3 when n >= 2.
T(3n-2,2n-2) = T(3n-3,2n-3) + 13 when n >= 4.

A384066 Limiting values for Cauchy-complete category table A384134.

Original entry on oeis.org

1, 2, 10, 39, 168

Offset: 0

Elijah Beregovsky, May 20 2025

This appears to be the absolute value of A165814.

Cf. A384134, A125697, A125701.

A384134(n,k) = a(n-k) if k >= (2/3)*n.

A384135 Number of Cauchy-complete categories with n morphisms.

Original entry on oeis.org

1, 2, 4, 11, 25, 63, 163, 451, 1311

Offset: 1

Elijah Beregovsky, May 20 2025

A Cauchy-complete (also called Karoubi-complete or idempotent-complete) category is one in which all idempotents split. In other words, in a Cauchy-complete category every arrow e:A→A such that e=e∘e has a retract, meaning there exists an object B and morphisms r:A→B and s:B→A such that s∘r=e but r∘s=1_B.

Geoff Cruttwell, Counting Finite Categories, presentation, (2018).
nLab, Cauchy complete category.
nLab, Karoubi envelope.

Cf. A125697.

A384190 Number of non-isomorphic AG-groupoids of order n.

Original entry on oeis.org

1, 3, 20, 331, 31913, 40104513, 643460323187

Offset: 1

Elijah Beregovsky, May 21 2025

A magma S is called an Abel-Grassmann or AG-groupoid (historically they were also called left almost semigroups, right modular groupoids and left invertive groupoids) if for all a,b,c in S (ab)c = (cb)a.

			For a(2) there are only 3 non-isomorphic AG-groupoids: the null semigroup, the semigroup formed by the set {0,1} under multiplication and the cyclic group Z2.

M. A. Kazim and M. Naseerudin, On almost semigroups, Alig. Bull. Math. 2, 1-7 (1972).

Marek Dančo, Mikoláš Janota, Michael Codish, and João Jorge Araújo, Complete Symmetry Breaking for Finite Models, arXiv:2502.10155 [cs.LO], 2025.
Marek Dančo, MACE-like model finder for algebraic structures.
Andreas Distler, Muhammad Shah, and Volker Sorge, Enumeration of AG-Groupoids, Davenport, J.H., Farmer, W.M., Urban, J., Rabe, F. (eds) Intelligent Computer Mathematics. CICM 2011. Lecture Notes in Computer Science, vol 6824. Springer, Berlin, Heidelberg.
Muhammad Iqbal, Imtiaz Ahmad, Muhammad Shah, and Muhammad Irfan Ali, On cyclic associative Abel-Grassman groupoids, arXiv:1510.01316 [math.GR], 2015.
Index entries for sequences related to groupoids

Cf. A001329 (magmas), A124506 (semigroups), A001426, A350875, A350874.

A383886 Number of 3-nilpotent semigroups, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Original entry on oeis.org

0, 0, 1, 8, 84, 2660, 609797, 1831687022, 52966239062973, 12417282095522918811, 26530703289252298687053072, 1008860098093547692911901804990610, 1378288413994605341053354105969660808031163, 36959929418354255758713676933402538920157765946956889, 14799968982226242179794503639146983952853044950740907666303436922

Offset: 1

Elijah Beregovsky, May 13 2025

nonn

A semigroup S is nilpotent if there exists a natural number r such that the set S^r of all products of r elements of S has size 1.
If r is the smallest such number, then S is said to have nilpotency degree r.
This sequence counts semigroups S that have an element e such that for all x,y,z in S x*y*z = e.
In 1976 Kleitman, Rothschild and Spencer gave an argument asserting that the proportion of 3-nilpotent semigroups, amongst all semigroups of order n, is asymptotically 1. Later opinion regards their argument as incomplete, and no satisfactory proof has been found.

H. Jürgensen, F. Migliorini, and J. Szép, Semigroups. Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest, 1991.

Andreas Distler and James D. Mitchell, The number of nilpotent semigroups of degree 3, arXiv:1201.3529 [math.CO], 2012.
Igor Dolinka, D. G. FitzGerald, and James D. Mitchell, Semirigidity and the enumeration of nilpotent semigroups of index three, arXiv:2411.00466 [math.CO], 2024.
Pierre A. Grillet, Counting Semigroups, Communications in Algebra, 43(2), 574-596, (2014).
D. J. Kleitman, B. R. Rothschild, and J. H. Spencer, The number of semigroups of order n, Proc. Amer. Math. Soc. 55 (1976), 227-232.
Index entries for sequences related to semigroups

Cf. A023814, A001423.

Cf. A383885, A383871.

a(n) = A383871(n)/2n! * (1+o(1)). See Grillet paper in Links.

A383885 Number of nonisomorphic 3-nilpotent semigroups of order n.

Original entry on oeis.org

0, 0, 1, 9, 118, 4671, 1199989, 3661522792, 105931872028455, 24834563582168716305, 53061406576514239124327751, 2017720196187069550262596208732035, 2756576827989210680367439732667802738773384, 73919858836708511517426763179873538289329852786253510, 29599937964452484359589007277447538854227891149791717673581110642

Offset: 1

Elijah Beregovsky, May 13 2025

nonn

A semigroup S is nilpotent if there exists a natural number r such that the set S^r of all products of r elements of S has size 1.
If r is the smallest such number, then S is said to have nilpotency degree r.
This sequence counts semigroups S that have an element e such that for all x,y,z in S x*y*z = e.
In 1976 Kleitman, Rothschild and Spencer gave an argument asserting that the proportion of 3-nilpotent semigroups, amongst all semigroups of order n, is asymptotically 1. Later opinion regards their argument as incomplete, and no satisfactory proof has been found.

H. Jürgensen, F. Migliorini, and J. Szép, Semigroups. Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest, 1991.

Andreas Distler and James D. Mitchell, The number of nilpotent semigroups of degree 3, arXiv:1201.3529 [math.CO], 2012.
Igor Dolinka, D. G. FitzGerald, and James D. Mitchell, Semirigidity and the enumeration of nilpotent semigroups of index three, arXiv:2411.00466 [math.CO], 2024.
Pierre A. Grillet, Counting Semigroups, Communications in Algebra, 43(2), 574-596, (2014).
D. J. Kleitman, B. R. Rothschild, and J. H. Spencer, The number of semigroups of order n, Proc. Amer. Math. Soc. 55 (1976), 227-232.
Index entries for sequences related to semigroups

Cf. A023814, A027851.

Cf. A383871, A383886.

a(n) = A383871(n)/n! * (1+o(1)). See Grillet paper in Links.

For exact formula see the Distler and Mitchell paper.

A383871 Number of labeled 3-nilpotent semigroups of order n.

Original entry on oeis.org

0, 0, 6, 180, 11720, 3089250, 5944080072, 147348275209800, 38430603831264883632, 90116197775746464859791750, 2118031078806486819496589635743440, 966490887282837500134221233339527160717340, 17165261053166610940029331024343115375665769316911576, 6444206974822296283920298148689544172139277283018112679406098010

Offset: 1

Elijah Beregovsky, May 13 2025

nonn

A semigroup S is nilpotent if there exists a natural number r such that the set S^r of all products of r elements of S has size 1.
If r is the smallest such number, then S is said to have nilpotency degree r.
This sequence counts semigroups S that have an element e such that for all x,y,z in S x*y*z = e.
In 1976 Kleitman, Rothschild and Spencer gave an argument asserting that the proportion of 3-nilpotent semigroups, amongst all semigroups of order n, is asymptotically 1. Later opinion regards their argument as incomplete, and no satisfactory proof has been found.

H. Jürgensen, F. Migliorini, and J. Szép, Semigroups. Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest, 1991.

Andreas Distler and James D. Mitchell, The number of nilpotent semigroups of degree 3, arXiv:1201.3529 [math.CO], 2012.
Igor Dolinka, D. G. FitzGerald, and James D. Mitchell, Semirigidity and the enumeration of nilpotent semigroups of index three, arXiv:2411.00466 [math.CO], 2024.
Pierre A. Grillet, Counting Semigroups, Communications in Algebra, 43(2), 574-596, (2014).
D. J. Kleitman, B. R. Rothschild, and J. H. Spencer, The number of semigroups of order n, Proc. Amer. Math. Soc. 55 (1976), 227-232.
Index entries for sequences related to semigroups

Cf. A023814, A023815.

Cf. A383885, A383886.

a(n) = Sum_{2 <= m <= b(n)} binomial(n,m) * m * Sum_{0 <= i <= m-1} (-1)^i * binomial(m-1,i) * (m-i)^((n-m)^2), where b(n) = floor(n + 1/2 - sqrt(n-3/4)).

A359205 Numbers that have at least two non-overlapping pairs of consecutive ones in their binary representation.

Original entry on oeis.org

15, 27, 30, 31, 47, 51, 54, 55, 59, 60, 61, 62, 63, 79, 91, 94, 95, 99, 102, 103, 107, 108, 109, 110, 111, 115, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 143, 155, 158, 159

Offset: 1

Elijah Beregovsky, Dec 23 2022

These are the numbers for which the smallest Hamming distance to a fibbinary number is larger than 1.

			27 is 11011 in binary, thus it is in the sequence.
14 is 1110 in binary. The pairs of consecutive ones overlap, so it is not in the sequence.

Wikipedia, Hamming distance
Wikipedia, Fibbinary number

Cf. A003714.

n=10;
a=Range[2^n];
fib=Select[a, BitAnd[#,2#]==0&];
nonadj=Complement[a,Union@@Outer[BitXor,fib,2^#&/@Range[n]]]

A330437 Length of trajectory of n under the map n -> n - 1 + n/gpf(n) or 0 if no fixed point is reached, where gpf(n) is the greatest prime factor of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 5, 4, 2, 1, 4, 1, 2, 4, 4, 3, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 3, 2, 2, 3, 1, 2, 2, 3, 2, 2, 1, 2, 1, 3, 2, 4, 3, 2, 1, 2, 2, 2, 1, 4, 1, 3, 2, 2, 2, 2, 1, 4, 2, 2, 1, 4, 2, 3, 2, 4, 1, 2, 2, 4, 4, 4, 3, 2, 1, 3, 2, 4

Offset: 1

Elijah Beregovsky, Feb 16 2020

nonn

The table of trajectories of n under is given in A329288.
All fixed points, besides 1, are prime.
Conjecture: every number appears in the sequence infinitely many times.
Conjecture: all terms are nonzero, i.e., every trajectory eventually reaches a prime.

			For n = 26 the trajectory is (26, 27, 35, 39, 41) so a(26) = 5.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006530 (greatest prime factor), A329288, A330704 (greedy inverse).

Cf. A270807, A120685, A331410.

g:= n -> n - 1 + n/max(numtheory:-factorset(n)):
f:= proc(n) option remember;
    if isprime(n) then 1 else 1+ procname(g(n)) fi
end proc:
f(1):= 1:
map(f, [$1..200]); # Robert Israel, May 01 2020

Clear[f,it,order,seq]; f[n_]:=f[n]=n-1+n/FactorInteger[n][[-1]][[1]]; it[k_,n_]:=it[k,n]=f[it[k,n-1]]; it[k_,1]=k; order[n_]:=order[n]=SelectFirst[Range[1,100], it[n,#]==it[n,#+1]&]; Print[order/@Range[1,100]];

apply( {a(n,c=1)=n>1&&while(nM. F. Hasler, Feb 19 2020

a(p) = 1 for any prime number p.

A329288 Table T(n,k) read by antidiagonals: T(n,k) = f(T(n,k)) starting with T(n,1)=n, where f(x) = x - 1 + x/gpf(x), that is, f(x) = A269304(x)-2.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 5, 5, 1, 2, 3, 5, 5, 6, 1, 2, 3, 5, 5, 7, 7, 1, 2, 3, 5, 5, 7, 7, 8, 1, 2, 3, 5, 5, 7, 7, 11, 9, 1, 2, 3, 5, 5, 7, 7, 11, 11, 10, 1, 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 1, 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 12

Offset: 1

Elijah Beregovsky, Feb 16 2020

If p=T(n,k0) is prime, then T(n,k) = p - 1 + p/p = p for k > k0. Thus, primes are fixed points of this map. The number of different terms in the n-th row is given by A330437.

			Table begins:
   1,  1,  1,  1,  1, ...
   2,  2,  2,  2,  2, ...
   3,  3,  3,  3,  3, ...
   4,  5,  5,  5,  5, ...
   5,  5,  5,  5,  5, ...
   6,  7,  7,  7,  7, ...
   7,  7,  7,  7,  7, ...
   8, 11, 11, 11, 11, ...
   9, 11, 11, 11, 11, ...
  10, 11, 11, 11, 11, ...
  11, 11, 11, 11, 11, ...
  12, 15, 17, 17, 17, ...
  13, 13, 13, 13, 13, ...
  14, 15, 17, 17, 17, ...

Cf. A006530 (greatest prime factor), A269304.

Clear[f,it,order,seq]; f[n_]:=f[n]=n-1+n/FactorInteger[n][[-1]][[1]]; it[k_,n_]:=it[k,n]=f[it[k,n-1]]; it[k_,1]=k; SetAttributes[f,Listable]; SetAttributes[it,Listable]; it[#,Range[10]]&/@Range[800]

cp's OEIS Frontend

User: Elijah Beregovsky

A384134 Triangle read by rows: T(n,k) is the number of Cauchy-complete categories with n morphisms and k objects.

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A384066 Limiting values for Cauchy-complete category table A384134.

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A384135 Number of Cauchy-complete categories with n morphisms.

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A384190 Number of non-isomorphic AG-groupoids of order n.

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A383886 Number of 3-nilpotent semigroups, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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A383885 Number of nonisomorphic 3-nilpotent semigroups of order n.

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A383871 Number of labeled 3-nilpotent semigroups of order n.

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A359205 Numbers that have at least two non-overlapping pairs of consecutive ones in their binary representation.

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A330437 Length of trajectory of n under the map n -> n - 1 + n/gpf(n) or 0 if no fixed point is reached, where gpf(n) is the greatest prime factor of n.

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