cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A096827 Number of antichains in divisor lattice D(n).

Original entry on oeis.org

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
Offset: 1

Views

Author

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

Keywords

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

Crossrefs

Cf. A096825, A096826, A097699.
Cf. A175177, A175178. - Artur Jasinski, Mar 01 2010
Cf. A000005, A008480, A253249, A285572 A285573.

Programs

  • 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 *)

Formula

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

Extensions

More terms from John W. Layman, Aug 20 2004

A175177 Conjectured number of numbers for which the iteration x -> phi(x) + 1 terminates at prime(n). Cardinality of rooted tree T_p (where p is n-th prime) in Karpenko's book.

Original entry on oeis.org

2, 3, 4, 9, 2, 31, 6, 4, 2, 2, 2, 11, 24, 41, 2, 2, 2, 57, 2, 2, 58, 2, 2, 6, 17, 4, 2, 2, 39, 67, 2, 2, 2, 2, 2, 2, 25, 4, 2, 2, 2, 158, 2, 61, 2, 2, 2, 2, 2, 2, 54, 2, 186, 2, 10, 2, 2, 2, 18, 8, 2, 2, 2, 2, 96, 2, 2, 18, 2, 6, 15, 2, 2, 2, 2, 2, 2, 44, 34, 6, 2, 16, 2, 105, 2, 2, 60, 5, 4, 2, 2, 2, 4
Offset: 1

Views

Author

Artur Jasinski, Mar 01 2010

Keywords

Examples

			a(3) = 4 because x = { 5, 8, 10, 12 } are the 4 numbers from which the iteration x -> phi(x) + 1 terminates at prime(3) = 5.
a(4) = 8 because x = { 7, 9, 14, 15, 16, 18, 20, 24, 30 } are the 9 numbers from which the iteration x -> phi(x) + 1 terminates at prime(4) = 7.

References

  • Richard K. Guy, Unsolved Problems in Number Theory, Third Edition, Springer, New York 2004. Chapter B41, Iterations of phi and sigma, page 148.
  • A. S. Karpenko, Lukasiewicz's Logics and Prime Numbers, (English translation), 2006. See Table 2 on p.125 ff.
  • A. S. Karpenko, Lukasiewicz's Logics and Prime Numbers, (Russian), 2000.

Links

Crossrefs

Cf. A039649, A039650, A039651, A039652, A096827, A175178.

Programs

  • PARI
    iterat(x) = {my(k,s); if ( isprime(x),return(x)); s=x;
for (k=1,1000000000,s=eulerphi(s)+1;if(isprime(s),return(s)));
return(s); }
check(y,endrange) = {my(count,start); count=0;
for(start=1,endrange,if(iterat(start)==y,count++;));
return(count); }
for (n=1,93,x=prime(n);print1(check(x,1000000),", "))
\\ Hugo Pfoertner, Sep 23 2017

Extensions

Name clarified by Hugo Pfoertner, Sep 23 2017

A175179 Primes for which value of CRT (Cardinality of rooted tree) is equal to 1.

Original entry on oeis.org

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
Offset: 1

Views

Author

Artur Jasinski, Mar 01 2010

Keywords

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).

Crossrefs

Cf. A096827, A175177, A175178.

A173883 a(n) = number of iterations in the sequence of classes of prime numbers for prime(n).

Original entry on oeis.org

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
Offset: 2

Views

Author

Artur Jasinski, Mar 01 2010

Keywords

References

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

Crossrefs

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

Extensions

Edited, corrected and extended by Arkadiusz Wesolowski, Jan 19 2013
Showing 1-4 of 4 results.

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