A343939
Number of n-chains of divisors of n.
Original entry on oeis.org
1, 3, 4, 15, 6, 49, 8, 165, 55, 121, 12, 1183, 14, 225, 256, 4845, 18, 3610, 20, 4851, 484, 529, 24, 73125, 351, 729, 4060, 12615, 30, 29791, 32, 435897, 1156, 1225, 1296, 494209, 38, 1521, 1600, 505981, 42, 79507, 44, 46575, 49726, 2209, 48
Offset: 1
The a(1) = 1 through a(5) = 6 chains:
(1) (1/1) (1/1/1) (1/1/1/1) (1/1/1/1/1)
(2/1) (3/1/1) (2/1/1/1) (5/1/1/1/1)
(2/2) (3/3/1) (2/2/1/1) (5/5/1/1/1)
(3/3/3) (2/2/2/1) (5/5/5/1/1)
(2/2/2/2) (5/5/5/5/1)
(4/1/1/1) (5/5/5/5/5)
(4/2/1/1)
(4/2/2/1)
(4/2/2/2)
(4/4/1/1)
(4/4/2/1)
(4/4/2/2)
(4/4/4/1)
(4/4/4/2)
(4/4/4/4)
Diagonal n = k - 1 of the array
A077592.
Chains of length n - 1 are counted by
A163767.
Diagonal n = k of the array
A334997.
The version counting all multisets of divisors (not just chains) is
A343935.
A067824(n) counts strict chains of divisors starting with n.
A074206(n) counts strict chains of divisors from n to 1.
A146291(n,k) counts divisors of n with k prime factors (with multiplicity).
A251683(n,k-1) counts strict k-chains of divisors from n to 1.
A253249(n) counts nonempty chains of divisors of n.
A334996(n,k) counts strict k-chains of divisors from n to 1.
A337255(n,k) counts strict k-chains of divisors starting with n.
A343658(n,k) counts k-multisets of divisors of n.
A343662(n,k) counts strict k-chains of divisors of n (row sums:
A337256).
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Table[Length[Select[Tuples[Divisors[n],n],OrderedQ[#]&&And@@Divisible@@@Reverse/@Partition[#,2,1]&]],{n,10}]
A343940
Sum of numbers of ways to choose a k-chain of divisors of n - k, for k = 0..n - 1.
Original entry on oeis.org
1, 2, 4, 7, 12, 19, 30, 45, 66, 95, 135, 187, 256, 346, 463, 613, 803, 1040, 1336, 1703, 2158, 2720, 3409, 4244, 5251, 6461, 7911, 9643, 11707, 14157, 17058, 20480, 24502, 29212, 34707, 41094, 48496, 57053, 66926, 78296, 91369, 106376, 123581, 143276, 165786
Offset: 1
The a(8) = 45 chains:
() (1) (1/1) (1/1/1) (1/1/1/1) (1/1/1/1/1) (1/1/1/1/1/1)
(7) (2/1) (5/1/1) (2/1/1/1) (3/1/1/1/1) (2/1/1/1/1/1)
(2/2) (5/5/1) (2/2/1/1) (3/3/1/1/1) (2/2/1/1/1/1)
(3/1) (5/5/5) (2/2/2/1) (3/3/3/1/1) (2/2/2/1/1/1)
(3/3) (2/2/2/2) (3/3/3/3/1) (2/2/2/2/1/1)
(6/1) (4/1/1/1) (3/3/3/3/3) (2/2/2/2/2/1)
(6/2) (4/2/1/1) (2/2/2/2/2/2)
(6/3) (4/2/2/1)
(6/6) (4/2/2/2)
(4/4/1/1)
(4/4/2/1) (1/1/1/1/1/1/1)
(4/4/2/2)
(4/4/4/1)
(4/4/4/2)
(4/4/4/4)
Antidiagonal sums of the array (or row sums of the triangle)
A334997.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1.
A146291 counts divisors of n with k prime factors (with multiplicity).
A251683 counts strict length k + 1 chains of divisors from n to 1.
A253249 counts nonempty chains of divisors of n.
A334996 counts strict length k chains of divisors from n to 1.
A337255 counts strict length k chains of divisors starting with n.
- version counting all multisets of divisors (not just chains)
A343658,
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Total/@Table[Length[Select[Tuples[Divisors[n-k],k],And@@Divisible@@@Partition[#,2,1]&]],{n,12},{k,0,n-1}]
A343936
Number of ways to choose a multiset of n divisors of n - 1.
Original entry on oeis.org
1, 2, 3, 10, 5, 56, 7, 120, 45, 220, 11, 4368, 13, 560, 680, 3876, 17, 26334, 19, 42504, 1771, 2024, 23, 2035800, 325, 3276, 3654, 201376, 29, 8347680, 31, 376992, 6545, 7140, 7770, 145008513, 37, 9880, 10660, 53524680, 41, 73629072, 43, 1712304, 1906884
Offset: 1
The a(1) = 1 through a(5) = 5 multisets:
{} {1} {1,1} {1,1,1} {1,1,1,1}
{2} {1,3} {1,1,2} {1,1,1,5}
{3,3} {1,1,4} {1,1,5,5}
{1,2,2} {1,5,5,5}
{1,2,4} {5,5,5,5}
{1,4,4}
{2,2,2}
{2,2,4}
{2,4,4}
{4,4,4}
The a(6) = 56 multisets:
11111 11136 11333 12236 13366 22266 23666
11112 11166 11336 12266 13666 22333 26666
11113 11222 11366 12333 16666 22336 33333
11116 11223 11666 12336 22222 22366 33336
11122 11226 12222 12366 22223 22666 33366
11123 11233 12223 12666 22226 23333 33666
11126 11236 12226 13333 22233 23336 36666
11133 11266 12233 13336 22236 23366 66666
The version for chains of divisors is
A163767.
Choosing n divisors of n gives
A343935.
A007318 counts k-sets of elements of {1..n}.
A009998 = n^k (as an array, offset 1).
A059481 counts k-multisets of elements of {1..n}.
A146291 counts divisors of n with k prime factors (with multiplicity).
A253249 counts nonempty chains of divisors of n.
Strict chains of divisors:
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A067824 counts strict chains of divisors starting with n.
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A074206 counts strict chains of divisors from n to 1.
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A251683 counts strict length k + 1 chains of divisors from n to 1.
-
A334996 counts strict length-k chains of divisors from n to 1.
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A337255 counts strict length-k chains of divisors starting with n.
-
A337256 counts strict chains of divisors of n.
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A343662 counts strict length-k chains of divisors.
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multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[multchoo[DivisorSigma[0,n],n-1],{n,50}]