A122977 -id:A122977 - cp's OEIS frontend

A122978 Number of sublattices of the division lattice of divisors of n.

1, 3, 3, 7, 3, 12, 3, 15, 7, 12, 3, 37, 3, 12, 12, 31, 3, 37, 3, 37, 12, 12, 3, 103, 7, 12, 15, 37, 3, 73, 3, 63, 12, 12, 12, 146, 3, 12, 12, 103, 3, 73, 3, 37, 37, 12, 3, 271, 7, 37, 12, 37, 3, 103, 12, 103, 12, 12, 3, 319, 3, 12, 37, 127, 12, 73, 3, 37, 12, 73, 3, 505, 3, 12, 37, 37

Offset: 1

Franklin T. Adams-Watters, Sep 21 2006

nonn

A division lattice is closed under GCD and LCM. Depends only on the prime signature of n.

Cf. A051839, A122977, A122981, A122982.

A163820 Number of permutations of the divisors of n that are greater than 1, in which consecutive elements are not coprime.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 6, 2, 2, 1, 36, 1, 2, 2, 24, 1, 36, 1, 36, 2, 2, 1, 1440, 2, 2, 6, 36, 1, 348, 1, 120, 2, 2, 2, 10560, 1, 2, 2, 1440, 1, 348, 1, 36, 36, 2, 1, 100800, 2, 36, 2, 36, 1, 1440, 2, 1440, 2, 2, 1, 2218560, 1, 2, 36, 720, 2, 348, 1, 36, 2, 348, 1, 9737280, 1, 2, 36, 36, 2, 348, 1, 100800, 24, 2, 1, 2218560, 2, 2, 2, 1440, 1, 2218560, 2, 36, 2, 2, 2, 10886400, 1, 36, 36, 10560

Offset: 1

Leroy Quet, Aug 04 2009

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(60) = a(90) since 60 = 2^2 * 3 * 5 and 90 = 2 * 3^2 * 5 both have prime signature (2,1,1). - Antti Karttunen, Oct 22 2017

As a consequence of the comment above, a(n) = a(A046523(n)). - David A. Corneth, Oct 22 2017

			The divisors of 12 that are > 1 are 2,3,4,6,12. In the permutations that are counted, 3 cannot be next to 2 or 4. However, a permutation that is among those counted is 6,2,4,12,3. The GCDs of adjacent pairs in this permutation are gcd(6,2)=2, gcd(2,4)=2, gcd(4,12)=4, gcd(12,3)=3. Note that all of these GCDs are > 1.

David A. Corneth, Table of n, a(n) for n = 1..359 (first 119 terms from Antti Karttunen)
Antti Karttunen, Scheme-program for computing this sequence
Index entries for sequences computed from exponents in factorization of n

Cf. also A046523, A114717, A119842, A122977, A122978, A293900, A293902.

Array[Count[Permutations@ Rest@ Divisors[#], ?(NoneTrue[Partition[#, 2, 1], CoprimeQ @@ # &] &)] - Boole[# == 1] &, 59] (* _Michael De Vlieger, Nov 04 2017 *)

a(p) = 1 for all primes p. a(p*q) = 2 for all pairs of (not necessarily distinct) primes p and q.
From Antti Karttunen, Oct 22 2017: (Start)
a(p^n) = A000142(n), for all primes p.
a(n) = A293900(n)*A293902(n).
(End)

Definition corrected by Leroy Quet, Aug 15 2009

Edited and extended by Max Alekseyev, Jun 13 2011

A122979 Number of distributive sublattices of the lattice of k-tuples less than the n-th partition (in Abramowitz and Stegun order), that include the maximum element.

Original entry on oeis.org

2, 4, 7, 8, 21, 45, 16, 58, 84, 200, 500, 32, 152, 293, 748, 1184, 3220

Offset: 1

Franklin T. Adams-Watters, Sep 21 2006

After a(18) - for partition [1^5] - the sequence continues ?, 64, 384, 938, 1238, 2520, 5591, ?, ?, ?, ?, ?, 128.

			For a(5), partition [2,1], the lattice consists of the 6 pairs (i,j) where 0<=i<=2 and 0<=j<=1, with (i,j) <= (i',j') iff i<=i' and j<=j'. {(2,1), (2,0), (0,1), (0,0)} is one distributive sublattice.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A122977, A122980, A122981, A036036, A074139.

A122980 Number of distributive sublattices of the lattice of k-tuples less than the n-th partition (in Mathematica order), that include the maximum element.

Original entry on oeis.org

2, 4, 7, 8, 21, 45, 16, 58, 84, 200, 500, 32, 152, 293, 748, 1184, 3220

Offset: 1

Franklin T. Adams-Watters, Sep 21 2006

After a(18) - for partition [1^5] - the sequence continues ?, 64, 384, 938, 2520, 1238, 5591, ?, ?, ?, ?, ?, 128.

			For a(5), partition [2,1], the lattice consists of the 6 pairs (i,j) where 0<=i<=2 and 0<=j<=1, with (i,j) <= (i',j') iff i<=i' and j<=j'. {(2,1), (2,0), (0,1), (0,0)} is one distributive sublattice.

Cf. A122977, A122979, A122982, A080577, A074139.

A074986 Number of nonempty subsets of {1,2,3,...,n} that are closed under both GCD and LCM.

Original entry on oeis.org

1, 3, 5, 9, 11, 18, 20, 28, 32, 39, 41, 62, 64, 71, 78, 94, 96, 117, 119, 140, 147, 154, 156, 214, 218, 225, 233, 254, 256, 301, 303, 335, 342, 349, 356, 440, 442, 449, 456, 514, 516, 561, 563, 584, 605, 612, 614, 766, 770, 791, 798, 819, 821, 879, 886, 944, 951

Offset: 1

John W. Layman, Oct 02 2002

nonn

I.e. number of division lattices with values from 1 to n. - Franklin T. Adams-Watters, Sep 21 2006

Sean A. Irvine, Table of n, a(n) for n = 1..200

Cf. A051839, A122977, A122978.

More terms from Franklin T. Adams-Watters, Sep 21 2006

cp's OEIS Frontend

A122978 Number of sublattices of the division lattice of divisors of n.

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A163820 Number of permutations of the divisors of n that are greater than 1, in which consecutive elements are not coprime.

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A122979 Number of distributive sublattices of the lattice of k-tuples less than the n-th partition (in Abramowitz and Stegun order), that include the maximum element.

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A122980 Number of distributive sublattices of the lattice of k-tuples less than the n-th partition (in Mathematica order), that include the maximum element.

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A074986 Number of nonempty subsets of {1,2,3,...,n} that are closed under both GCD and LCM.

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