A162247 -id:A162247 - cp's OEIS frontend

A344088 Flattened tetrangle of reversed strict integer partitions sorted first by sum, then colexicographically.

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

Offset: 0

Gus Wiseman, May 12 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (12)(3)
  4: (13)(4)
  5: (23)(14)(5)
  6: (123)(24)(15)(6)
  7: (124)(34)(25)(16)(7)
  8: (134)(125)(35)(26)(17)(8)
  9: (234)(135)(45)(126)(36)(27)(18)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
The non-strict version is A080576.
Taking lex instead of colex gives A246688 (non-reversed: A344086).
The non-reversed version is A344087.
Taking revlex instead of colex gives A344089 (non-reversed: A118457).
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344091.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Table[Sort[Reverse/@Select[IntegerPartitions[n],UnsameQ@@#&],colex],{n,0,10}]

A355030 a(n) is the number of possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 11, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 11, 5, 2, 1, 11, 2

Offset: 1

Amiram Eldar, Jun 16 2022

nonn

First differs from A305254 at n = 40, from A001055 and A252665 at n = 36, from A218320 at n = 32 and from A317791, A318559 and A326334 at n = 30.

			a(2) = 1 since numbers with 2 divisors are primes, i.e., numbers k with the single value Omega(k) = 1.
a(4) = 2 since numbers with 4 divisors are either of the following 2 forms: p1 * p2 with p1 and p2 being distinct primes, or of the form p^3 with p prime.
a(8) = 3 since numbers with 8 divisors are either of the following 3 forms: p1 * p2 * p3 with p1, p2 and p3 being distinct primes, p1 * p2^3, or p1^7.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Row lengths of A355029.
Cf. A000005, A001222, A118914, A162247, A355027.
Cf. A001055, A218320, A305254, A317791, A318559, A326334.

Table[Length[Union[Total[#-1]& /@ f[n]]], {n, 1, 100}] (* using the function f by T. D. Noe at A162247 *)

a(n) <= A001055(n).
a(p) = 1 for p prime.
a(A355031(n)) = n.

A355029 Irregular table read by rows: the n-th row gives the possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 3, 5, 6, 3, 4, 7, 4, 8, 5, 9, 10, 4, 5, 6, 11, 12, 7, 13, 6, 14, 4, 5, 6, 8, 15, 16, 5, 7, 9, 17, 18, 6, 7, 10, 19, 8, 20, 11, 21, 22, 5, 6, 7, 8, 9, 12, 23, 8, 24, 13, 25, 6, 10, 26, 8, 9, 14, 27, 28, 7, 9, 11, 15, 29, 30, 5, 6, 7, 9, 10, 16, 31

Offset: 1

Amiram Eldar, Jun 16 2022

The n-th row begins with A059975(n) and ends with n-1.

			Table begins:
  0;
  1;
  2;
  2, 3;
  4;
  3, 5;
  6;
  3, 4, 7;
  4, 8;
  5, 9;
  ...
Numbers k with 4 divisors are either of the form p1 * p2 with p1 and p2 being distinct primes, or of the form p^3 with p prime. The corresponding numbers of prime divisors (counted with multiplicity) are 2 and 3, respectively. Therefore, the 4th row is {2, 3}.

Amiram Eldar, Table of n, a(n) for n = 1..6758 (rows 1..1000, flattened)

Cf. A000005, A001222, A059975, A118914, A162247, A355026, A355030 (row lengths).

Table[Union[Total[#-1]& /@ f[n]], {n, 1, 32}] // Flatten (* using the function f by T. D. Noe at A162247 *)

A355031 a(n) is the least number k such that A355030(k) = n, or -1 if no such k exists.

Original entry on oeis.org

1, 4, 8, 12, 16, 40, 24, 36, 90, 126, 48, 112, 546, 72, 108, 96, 160, 352, 168, 120, 256, 2475, 144, 588, 300, 320, 216, 448, 1216, 240, 810, 420, 288, 1040, 384, 660, 360, 640, 432, 1408, 540, 504, 480, 600, 648, 1176, 792, 672, 1500, 576, 2000, 900, 1824, 1248

Offset: 1

Amiram Eldar, Jun 16 2022

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001222, A162247, A355029, A355030.

m[n_] := Length[Union[Total[#-1]& /@ f[n]]]; seq[len_, max_] := Module[{s = Table[0, {len}], c = 0, n = 1, k}, While[c < len && n < max, k = m[n]; If[k <= len && s[[k]] == 0, c++; s[[k]] = n]; n++]; s]; seq[60, 10^4] (* using the function f by T. D. Noe at A162247 *)

A301598 Number of thrice-factorizations of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 1, 10, 4, 4, 1, 16, 1, 4, 4, 34, 1, 16, 1, 16, 4, 4, 1, 54, 4, 4, 10, 16, 1, 22, 1, 80, 4, 4, 4, 78, 1, 4, 4, 54, 1, 22, 1, 16, 16, 4, 1, 181, 4, 16, 4, 16, 1, 54, 4, 54, 4, 4, 1, 102, 1, 4, 16, 254, 4, 22, 1, 16, 4, 22, 1, 272, 1, 4, 16, 16

Offset: 1

Gus Wiseman, Mar 24 2018

nonn

A thrice-factorization of n is a choice of a twice-factorization of each factor in a factorization of n. Thrice-factorizations correspond to intervals in the lattice form of the multiorder of integer factorizations.

			The a(12) = 16 thrice-factorizations:
((2))*((2))*((3)), ((2))*((2)*(3)), ((3))*((2)*(2)), ((2)*(2)*(3)),
((2))*((2*3)), ((2)*(2*3)),
((2))*((6)), ((2)*(6)),
((3))*((2*2)), ((3)*(2*2)),
((3))*((4)), ((3)*(4)),
((2*2*3)),
((2*6)),
((3*4)),
((12)).

Gus Wiseman, The (2*2*3*3) component of the lattice form of the multiorder of integer factorizations.
Gus Wiseman, The (2*2*3*3) component, version 2.

Cf. A001055, A007716, A050336, A050338, A063834, A162247, A269134, A281113, A281116, A301595, A301598, A301706.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
twifacs[n_]:=Join@@Table[Tuples[facs/@f],{f,facs[n]}];
thrifacs[n_]:=Join@@Table[Tuples[twifacs/@f],{f,facs[n]}];
Table[Length[thrifacs[n]],{n,15}]

Dirichlet g.f.: Product_{n > 1} 1/(1 - A281113(n)/n^s).

A317534 Numbers k such that the poset of factorizations of k, ordered by refinement, is not a lattice.

Original entry on oeis.org

24, 32, 40, 48, 54, 56, 60, 64, 72, 80, 84, 88, 90, 96, 104, 108, 112, 120, 126, 128, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 198, 200, 204, 208, 216, 220, 224, 228, 232, 234, 240, 243, 248, 250, 252, 256, 260, 264, 270

Offset: 1

Gus Wiseman, Jul 30 2018

nonn

Includes 2^k for all k > 4.

Conjecture: Let S be the set of all numbers whose prime signature is either {1,3}, {5}, or {1,1,2}. Then the sequence consists of all multiples of elements of S. - David A. Corneth, Jul 31 2018.

			In the poset of factorizations of 24, the factorizations (2*2*6) and (2*3*4) have two least-upper bounds, namely (2*12) and (4*6), so this poset is not a lattice.

R. P Stanley, Enumerative Combinatorics Vol. 1, Sec. 3.3.

Wikipedia, Lattice (order)

Cf. A001055, A007716, A025487, A045778, A065036, A162247, A265947, A281113, A317142, A317144, A317145, A317146.

A319239 Positions of nonzero terms in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 13, 16, 17, 19, 23, 24, 27, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 47, 53, 54, 56, 59, 60, 61, 64, 66, 67, 70, 71, 73, 78, 79, 81, 83, 84, 88, 89, 90, 96, 97, 100, 101, 102, 103, 104, 105, 107, 109, 110, 113, 114, 120, 125, 126, 127, 128

Offset: 1

Gus Wiseman, Sep 15 2018

nonn

Complement of A319240.

Cf. A001055, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319237.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Join@@Position[Table[Sum[(-1)^Length[f],{f,facs[n]}],{n,100}],_Integer?(Abs[#]>0&)]

A320835 a(n) = Sum (-1)^k where the sum is over all multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or factorizations of A181821(n).

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 1, 1, 1, 1, -1, 1, 1, 0, 0, 1, -1, 0, 2, 1, 1, 1, -2, 0, 1, 0, 0, 0, 2, 0, -2, -2, -1, 1, -1, -2, 3, -1, 1, -2, -3, -2, 3, 0, -3, 1, -4, -5, 1, -1, -2, -1, 5, -5, 1, -3, 1, -1, -5, -4, 5, 1, -1, -9, -2, -1, -6, -1, -3, -2, 7, -7, -8, -2, -2

Offset: 1

Gus Wiseman, Oct 21 2018

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A001055, A001222, A007716, A045778, A114592, A162247, A181821, A305936, A316441, A318284, A319237, A319238, A320836.

with(numtheory):
b:= proc(n, k) option remember; `if`(n>k, 0, -1)+`if`(isprime(n), 0,
      -add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> `if`(n=1, 1, b(((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
         sort(map(i-> pi(i[1])$i[2], ifactors(n)[2]), `>`)))$2)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 23 2018

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[(-1)^(Length[m]-1),{m,mps[nrmptn[n]]}],{n,30}]

a(n) = A316441(A181821(n)).

More terms from Alois P. Heinz, Oct 21 2018

A320836 a(n) = Sum (-1)^k where the sum is over all strict multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or strict factorizations of A181821(n).

Original entry on oeis.org

1, -1, -1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, 0, -2, -1, 0, -2, 0, -2, -1, -1, -1, -4, -1, -1, -1, -3, 0, -3, 0, -2, -4, -1, -1, -6, -2, -3, -2, -2, 0, -6, -2, -4, -1, -1, 0, -5, 0, -1, -3, -9, -2, -3, 0, -2, -1, -3, 0, -7, 0

Offset: 1

Gus Wiseman, Oct 21 2018

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A001055, A001222, A007716, A045778, A114592, A162247, A181821, A305936, A316441, A318284, A319237, A319238, A320835.

with(numtheory):
b:= proc(n, k) option remember; `if`(n>k, 0, -1)+`if`(isprime(n), 0,
      -add(`if`(d>k, 0, b(n/d, d-1)), d=divisors(n) minus {1, n}))
    end:
a:= n-> `if`(n=1, 1, b(((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
         sort(map(i-> pi(i[1])$i[2], ifactors(n)[2]), `>`)))$2)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 23 2018

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[(-1)^Length[m],{m,Select[mps[nrmptn[n]],UnsameQ@@#&]}],{n,30}]

a(n) = A114592(A181821(n)).

More terms from Alois P. Heinz, Oct 21 2018

A190892 Numbers that can be written as ab = cd*e, where a, b, c, d, and e are distinct composite numbers.

Original entry on oeis.org

192, 216, 240, 288, 320, 336, 360, 384, 432, 448, 480, 504, 528, 540, 560, 576, 600, 624, 640, 648, 672, 704, 720, 756, 768, 792, 800, 810, 816, 832, 840, 864, 880, 896, 900, 912, 936, 960, 972, 1000, 1008, 1024, 1040, 1056, 1080, 1088, 1104, 1120, 1134

Offset: 1

T. D. Noe, May 23 2011

nonn

Similar to A175340, but without the requirement that the composite numbers be consecutive. In this case, the k in A175340 can be taken to be 2.

Almost all numbers are in this sequence. Its complement has density O(n (log log n)^4/log n). - Charles R Greathouse IV, May 23 2011

			192 = 12*16 = 4*6*8.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A162247 (all factorizations of numbers), A175340.

A339245 is a subsequence.

cp's OEIS Frontend

A344088 Flattened tetrangle of reversed strict integer partitions sorted first by sum, then colexicographically.

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A355030 a(n) is the number of possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

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A355029 Irregular table read by rows: the n-th row gives the possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

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A355031 a(n) is the least number k such that A355030(k) = n, or -1 if no such k exists.

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A301598 Number of thrice-factorizations of n.

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A317534 Numbers k such that the poset of factorizations of k, ordered by refinement, is not a lattice.

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A319239 Positions of nonzero terms in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).

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A320835 a(n) = Sum (-1)^k where the sum is over all multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or factorizations of A181821(n).

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A320836 a(n) = Sum (-1)^k where the sum is over all strict multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or strict factorizations of A181821(n).

A190892 Numbers that can be written as ab = cd*e, where a, b, c, d, and e are distinct composite numbers.