A175177 -id:A175177

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A096827

Number of antichains in divisor lattice D(n).

+10
23

2, 3, 3, 4, 3, 6, 3, 5, 4, 6, 3, 10, 3, 6, 6, 6, 3, 10, 3, 10, 6, 6, 3, 15, 4, 6, 5, 10, 3, 20, 3, 7, 6, 6, 6, 20, 3, 6, 6, 15, 3, 20, 3, 10, 10, 6, 3, 21, 4, 10, 6, 10, 3, 15, 6, 15, 6, 6, 3, 50, 3, 6, 10, 8, 6, 20, 3, 10, 6, 20, 3, 35, 3, 6, 10, 10, 6, 20, 3, 21, 6, 6, 3, 50, 6, 6, 6, 15, 3, 50, 6

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The divisor lattice D(n) is the lattice of the divisors of the natural number n.

The empty set is counted as an antichain in D(n).

a(n) = gamma(n+1) where gamma is degree of cardinal completeness of Łukasiewicz n-valued logic. - Artur Jasinski, Mar 01 2010

REFERENCES

Alexander S. Karpenko, Lukasiewicz's Logics and Prime Numbers, Luniver Press, Beckington, 2006. See Table I p. 113.

LINKS

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..990

Index entries for sequences related to Łukasiewicz

FORMULA

a(n) = A285573(n) + 1. - Gus Wiseman, Aug 24 2018

MATHEMATICA

nn=200;

stableSets[u_, Q_]:=If[Length[u]===0, {{}}, With[{w=First[u]}, Join[stableSets[DeleteCases[u, w], Q], Prepend[#, w]&/@stableSets[DeleteCases[u, r_/; r===w||Q[r, w]||Q[w, r]], Q]]]];

Table[Length[stableSets[Divisors[n], Divisible]], {n, nn}] (* Gus Wiseman, Aug 24 2018 *)

CROSSREFS

Cf. A096825, A096826, A097699.

Cf. A175177, A175178. - Artur Jasinski, Mar 01 2010

Cf. A000005, A008480, A253249, A285572 A285573.

KEYWORD

nonn

AUTHOR

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 17 2004

EXTENSIONS

More terms from John W. Layman, Aug 20 2004

STATUS

approved

A175178

a(n)=Values of cardinality of rooted trees CRT for successive primes.

+10
4

1, 1, 1, 2, 1, 5, 1, 1, 1, 1, 1, 2, 4, 6, 1, 1, 2, 9, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 5, 6, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 16, 1, 9, 2, 1, 1, 1, 1, 1, 7, 1, 19, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 6, 3, 1, 1, 2, 1, 11, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

REFERENCES

Karpenko A.S. 2006. Lukasiewicz's Logics and Prime Numbers (English translation).

Karpenko A.S. 2000. Lukasiewicz's Logics and Prime Numbers (Russian).

LINKS

Table of n, a(n) for n=1..102.

CROSSREFS

A058340, A058340, A096827, A175177.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Mar 01 2010

STATUS

approved

A175179

Primes for which value of CRT (Cardinality of rooted tree) is equal to 1.

+10
1

2, 3, 5, 11, 17, 19, 23, 29, 31, 47, 53, 67, 71, 79, 83, 89, 97, 101, 103, 107, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 191, 199

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Primes p = Prime(x) such that A175178(x)=1.

REFERENCES

Karpenko A.S. 2006. Lukasiewicz's Logics and Prime Numbers (English translation).

Karpenko A.S. 2000. Lukasiewicz's Logics and Prime Numbers (Russian).

LINKS

Table of n, a(n) for n=1..32.

CROSSREFS

Cf. A096827, A175177, A175178.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Mar 01 2010

STATUS

approved

A173883

a(n) = number of iterations in the sequence of classes of prime numbers for prime(n).

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,8

REFERENCES

Alexander S. Karpenko, Lukasiewicz's Logics and Prime Numbers, Luniver Press, Beckington, 2006, pp. 98-102.

LINKS

Table of n, a(n) for n=2..89.

CROSSREFS

Cf. A058340, A096827, A175177, A175178, A175179.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Mar 01 2010

EXTENSIONS

Edited, corrected and extended by Arkadiusz Wesolowski, Jan 19 2013

STATUS

approved

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