A067824 -id:A067824 - cp's OEIS frontend

A378223 Inverse Möbius transform of A345182.

1, 1, 2, 2, 2, 4, 2, 4, 4, 4, 2, 10, 2, 4, 6, 8, 2, 12, 2, 10, 6, 4, 2, 24, 4, 4, 8, 10, 2, 20, 2, 16, 6, 4, 6, 36, 2, 4, 6, 24, 2, 20, 2, 10, 16, 4, 2, 56, 4, 12, 6, 10, 2, 32, 6, 24, 6, 4, 2, 62, 2, 4, 16, 32, 6, 20, 2, 10, 6, 20, 2, 100, 2, 4, 16, 10, 6, 20, 2, 56, 16, 4, 2, 62, 6, 4, 6, 24, 2, 72, 6, 10, 6, 4, 6

Offset: 1

Antti Karttunen, Nov 25 2024

nonn

Apparently the Dirichlet convolution of A002131 and A323910. - Antti Karttunen, Nov 30 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A002131, A323910, A345182, A378224 (Dirichlet inverse).
Cf. also A067824.
Odd bisection is not equal to A278223.

memoA345182 = Map();
A345182(n) = if(n<=2, n%2, my(v); if(mapisdefined(memoA345182,n,&v), v, v = sumdiv(n,d,if(dA345182(d),0)); mapput(memoA345182,n,v); (v)));
A378223(n) = sumdiv(n,d,A345182(d));

up_to = 20000;
A378223list(up_to_n) = { my(v=vector(up_to_n)); v[1] = 1; v[2] = 0; for(n=3,up_to_n,v[n] = 1+sumdiv(n,d,(dA378223list(up_to);
A378223(n) = v378223[n];

a(n) = Sum_{d|n} A345182(d).

For n > 2, a(n) = 2*A345182(n).

A124433 Irregular array {a(n,m)} read by rows where (sum{n>=1} sum{m=1 to A001222(n)+1} a(n,m)*y^m/n^x) = 1/(zeta(x)-1+1/y) for all x and y where the double sum converges.

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Dec 15 2006

Row n has A001222(n)+1 terms. The polynomial P_n(y) = (sum{m=1 to A001222(n)+1} a(n,m)*y^m) is a generalization of the Mobius (Moebius) function, where P_n(1) = A008683(n).
From Gus Wiseman, Aug 24 2020: (Start)
Up to sign, also the number of strict length-k chains of divisors from n to 1, 1 <= k <= 1 + A001222(n). For example, row n = 36 counts the following chains (empty column indicated by dot):
  .  36/1  36/2/1   36/4/2/1   36/12/4/2/1
           36/3/1   36/6/2/1   36/12/6/2/1
           36/4/1   36/6/3/1   36/12/6/3/1
           36/6/1   36/9/3/1   36/18/6/2/1
           36/9/1   36/12/2/1  36/18/6/3/1
           36/12/1  36/12/3/1  36/18/9/3/1
           36/18/1  36/12/4/1
                    36/12/6/1
                    36/18/2/1
                    36/18/3/1
                    36/18/6/1
                    36/18/9/1
(End)

			1/(zeta(x) - 1 + 1/y) = y - y^2/2^x - y^2/3^x + ( - y^2 + y^3)/4^x - y^2/5^x + ( - y^2 + 2y^3)/6^x - y^2/7^x + ...
From _Gus Wiseman_, Aug 24 2020: (Start)
The sequence of rows begins:
     1: 1              16: 0 -1 3 -3 1     31: 0 -1
     2: 0 -1           17: 0 -1            32: 0 -1 4 -6 4 -1
     3: 0 -1           18: 0 -1 4 -3       33: 0 -1 2
     4: 0 -1 1         19: 0 -1            34: 0 -1 2
     5: 0 -1           20: 0 -1 4 -3       35: 0 -1 2
     6: 0 -1 2         21: 0 -1 2          36: 0 -1 7 -12 6
     7: 0 -1           22: 0 -1 2          37: 0 -1
     8: 0 -1 2 -1      23: 0 -1            38: 0 -1 2
     9: 0 -1 1         24: 0 -1 6 -9 4     39: 0 -1 2
    10: 0 -1 2         25: 0 -1 1          40: 0 -1 6 -9 4
    11: 0 -1           26: 0 -1 2          41: 0 -1
    12: 0 -1 4 -3      27: 0 -1 2 -1       42: 0 -1 6 -6
    13: 0 -1           28: 0 -1 4 -3       43: 0 -1
    14: 0 -1 2         29: 0 -1            44: 0 -1 4 -3
    15: 0 -1 2         30: 0 -1 6 -6       45: 0 -1 4 -3
(End)

Mohammad K. Azarian, A Double Sum, Problem 440, College Mathematics Journal, Vol. 21, No. 5, Nov. 1990, p. 424. Solution published in Vol. 22. No. 5, Nov. 1991, pp. 448-449.

A008480 gives rows ends (up to sign).
A008683 gives row sums (the Moebius function).
A073093 gives row lengths.
A074206 gives unsigned row sums.
A097805 is the restriction to powers of 2 (up to sign).
A251683 is the unsigned version with zeros removed.
A334996 is the unsigned version (except with a(1) = 0).
A334997 is an unsigned non-strict version.
A337107 is the restriction to factorial numbers.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1.
A122651 counts strict chains of divisors summing to n.
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts strict chains of divisors.
A337105 counts strict chains of divisors from n! to 1.
Cf. A000005, A001221, A002033, A124010, A167865.

f[l_List] := Block[{n = Length[l] + 1, c},c = Plus @@ Last /@ FactorInteger[n];Append[l, Prepend[ -Plus @@ Pick[PadRight[ #, c] & /@ l, Mod[n, Range[n - 1]], 0],0]]];Nest[f, {{1}}, 34] // Flatten(* Ray Chandler, Feb 13 2007 *)
chnsc[n_]:=If[n==1,{{}},Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,DeleteCases[Divisors[n],1|n]}],{n}]];
Table[(-1)^k*Length[Select[chnsc[n],Length[#]==k&]],{n,30},{k,0,PrimeOmega[n]}] (* Gus Wiseman, Aug 24 2020 *)

a(1,1)=1. a(n,1) = 0 for n>=2. a(n,m+1) = -sum{k|n,k < n} a(k,m), where, for the purpose of this sum, a(k,m) = 0 if m > A001222(k)+1.

Extended by Ray Chandler, Feb 13 2007

A317100 Number of series-reduced planted achiral trees with n leaves spanning an initial interval of positive integers.

Original entry on oeis.org

1, 3, 5, 12, 17, 41, 65, 144, 262, 533, 1025, 2110, 4097, 8261, 16407, 32928, 65537, 131384, 262145, 524854, 1048647, 2098181, 4194305, 8390924, 16777234, 33558533, 67109132, 134226070, 268435457, 536887919, 1073741825, 2147516736, 4294968327, 8590000133

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

In these trees, achiral means that all branches directly under any given node that is not a leaf or a cover of leaves are equal, and series-reduced means that every node that is not a leaf or a cover of leaves has at least two branches.

			The a(4) = 12 trees:
  (1111), ((11)(11)), (((1)(1))((1)(1))), ((1)(1)(1)(1)),
  (1222),
  (1122), ((12)(12)),
  (1112),
  (1233),
  (1223),
  (1123),
  (1234).

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A001678, A003238, A052409, A052410, A067824, A167865, A168532, A214577, A289078, A294336, A316782, A317099.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
b[n_]:=1+Sum[b[n/d],{d,Rest[Divisors[n]]}];
a[n_]:=Sum[b[GCD@@Length/@Split[ptn]],{ptn,allnorm[n]}];
Array[a,10]

seq(n)={my(v=vector(n)); for(n=1, n, v[n]=2^(n-1) + sumdiv(n, d, v[d])); v} \\ Andrew Howroyd, Aug 19 2018

a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 07 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 19 2018

A342494 Number of compositions of n with strictly decreasing first quotients.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 15, 21, 30, 39, 50, 65, 82, 103, 129, 160, 196, 240, 293, 352, 422, 500, 593, 706, 832, 974, 1138, 1324, 1534, 1783, 2054, 2362, 2712, 3108, 3552, 4051, 4606, 5232, 5935, 6713, 7573, 8536, 9597, 10773, 12085, 13534, 15119, 16874, 18809

Offset: 0

Gus Wiseman, Mar 17 2021

nonn

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The composition (1,2,3,4,2) has first quotients (2,3/2,4/3,1/2) so is counted under a(12).
The a(1) = 1 through a(6) = 12 compositions:
  (1)  (2)    (3)    (4)      (5)      (6)
       (1,1)  (1,2)  (1,3)    (1,4)    (1,5)
              (2,1)  (2,2)    (2,3)    (2,4)
                     (3,1)    (3,2)    (3,3)
                     (1,2,1)  (4,1)    (4,2)
                              (1,2,2)  (5,1)
                              (1,3,1)  (1,2,3)
                              (2,2,1)  (1,3,2)
                                       (1,4,1)
                                       (2,3,1)
                                       (3,2,1)
                                       (1,2,2,1)

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The weakly decreasing version is A069916.
The version for differences instead of quotients is A325548.
The strictly increasing version is A342493.
The unordered version is A342499, ranked by A342525.
The strict unordered version is A342518.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A274199 counts compositions with all adjacent parts x < 2y.
Cf. A003242, A008965, A048004, A059966, A067824, A167606, A253249, A318991, A318992, A342527, A342528.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Greater@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]

a(21)-a(49) from Alois P. Heinz, Mar 18 2021

A342526 Heinz numbers of integer partitions with weakly decreasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87

Offset: 1

Gus Wiseman, Mar 23 2021

nonn

Also called log-concave-down partitions.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The prime indices of 294 are {1,2,4,4}, with first quotients (2,2,1), so 294 is in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   72: {1,1,1,2,2}
   76: {1,1,8}
   78: {1,2,6}
   80: {1,1,1,1,3}
   84: {1,1,2,4}

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version counting strict divisor chains is A057567.
For multiplicities (prime signature) instead of quotients we have A242031.
For differences instead of quotients we have A325361 (count: A320466).
These partitions are counted by A342513 (strict: A342519, ordered: A069916).
The weakly increasing version is A342523.
The strictly decreasing version is A342525.
A000929 counts partitions with all adjacent parts x >= 2y.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A002843 counts compositions with all adjacent parts x <= 2y.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A318991/A318992 rank reversed partitions with/without integer quotients.
Cf. A048767, A056239, A067824, A112798, A238710, A253249, A325351, A325352, A325405, A334997, A342086, A342191.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],GreaterEqual@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]

A343371 a(n) = 1 + Sum_{d|n, d < n} a(d - 1).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 5, 4, 6, 2, 9, 2, 8, 7, 7, 2, 12, 2, 12, 9, 9, 2, 13, 5, 12, 9, 12, 2, 22, 2, 14, 10, 10, 10, 18, 2, 15, 13, 16, 2, 26, 2, 20, 20, 12, 2, 22, 7, 23, 11, 19, 2, 26, 11, 23, 16, 15, 2, 30, 2, 25, 26, 16, 14, 36, 2, 22, 13, 27, 2, 32, 2, 21, 28

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A003238, A067824, A068336, A167865.

a:= proc(n) option remember;
      1+add(a(d-1), d=numtheory[divisors](n) minus {n})
    end:
seq(a(n), n=0..75);  # Alois P. Heinz, Apr 12 2021

a[n_] := a[n] = 1 + Sum[If[d < n, a[d - 1], 0], {d, Divisors[n]}]; Table[a[n], {n, 0, 75}]
nmax = 75; A[] = 0; Do[A[x] = 1/(1 - x) + Sum[x^k A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) //Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^2 * A(x^2) + x^3 * A(x^3) + x^4 * A(x^4) + ...

A343661 Sum of numbers of y-multisets of divisors of x for each x >= 1, y >= 0, x + y = n.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 46, 70, 105, 155, 223, 316, 443, 619, 865, 1210, 1690, 2354, 3263, 4497, 6157, 8368, 11280, 15078, 19989, 26296, 34356, 44626, 57693, 74321, 95503, 122535, 157101, 201377, 258155, 330994, 424398, 544035, 696995, 892104, 1140298, 1455080

Offset: 1

Gus Wiseman, Apr 30 2021

nonn

			The a(5) = 12 multisets of divisors:
  {1,1,1,1}  {1,1,1}  {1,1}  {1}  {}
             {1,1,2}  {1,3}  {2}
             {1,2,2}  {3,3}  {4}
             {2,2,2}

Antidiagonal sums of the array A343658 (or row sums of the triangle).
Dominates A343657.
A000005 counts divisors.
A007318 counts k-sets of elements of {1..n}.
A059481 counts k-multisets of elements of {1..n}.
A343656 counts divisors of powers.
Cf. A000169, A000312, A009998, A062319, A067824, A143773, A146291, A176029, A184389, A285572, A326077, A327527, A334996, A343652, A343657.

multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[Sum[multchoo[DivisorSigma[0,k],n-k],{k,n}],{n,10}]

a(n) = Sum_{k=1..n} binomial(sigma(k) + n - k - 1, n - k).

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

Gus Wiseman, May 05 2021

nonn

			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.
A000005(n) counts divisors of n.
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).
Cf. A062319, A066959, A143773, A176029, A327527, A343656.

Table[Length[Select[Tuples[Divisors[n],n],OrderedQ[#]&&And@@Divisible@@@Reverse/@Partition[#,2,1]&]],{n,10}]

A378649 Möbius transform of A378647.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 8, 12, 6, 8, 8, 16, 10, 18, 10, 12, 10, 22, 20, 20, 12, 18, 12, 28, 20, 30, 16, 20, 16, 24, 32, 36, 18, 24, 26, 40, 24, 42, 20, 24, 22, 46, 48, 42, 20, 32, 24, 52, 30, 40, 32, 36, 28, 58, 64, 60, 30, 36, 32, 48, 32, 66, 32, 44, 28, 70, 88, 72, 36, 40, 36, 60, 36, 78, 60, 54, 40, 82

Offset: 1

Antti Karttunen, Dec 03 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A005101, A008683, A067824, A103977, A263837, A378644, A378647 (inverse Möbius transform).

A378649(n) = sumdiv(n,d,moebius(d)*A378647(n/d));

a(n) = Sum_{d|n} A008683(d)*A378647(n/d):
a(n) = (A378644(n)+A378647(n)) / 2.
a(n) = A000010(n) = phi(n) only when n is a non-abundant number (A263837), and apparently never when n is abundant, A005101.

A107748 Number of monic divisors of x^n - 1 with coefficients in {0,1,-1}.

Original entry on oeis.org

2, 4, 4, 8, 4, 14, 4, 16, 8, 14, 4, 48, 4, 14, 14, 32, 4, 50, 4, 48, 14, 14, 4, 162, 8, 14, 16, 48, 4, 136, 4, 64, 14, 14, 14, 286, 4, 14, 14, 160, 4, 136, 4, 48, 48, 14, 4, 550, 8, 50, 14, 48, 4, 186, 14, 164, 14, 14, 4, 1124, 4, 14, 48, 128, 14, 136, 4, 48, 14, 136

Offset: 1

W. Edwin Clark, Jun 11 2005

nonn

Multiply by 2 to get all that have coefficients in {0,1,-1}.
Note that many of these are equal to 2^tau(n), where tau(n) is the number of positive divisors of n = number of irreducible factors of x^n - 1. This is connected with the fact that for small values of n the coefficients of the n-th cyclotomic polynomial belong to {0,1,-1}.
From Robert Israel, Aug 24 2017: (Start)
Each of these polynomials is a product of distinct cyclotomic polynomials C_k(x) for k dividing n.
a(n) <= 2^tau(n).
If n is prime then a(n)=4. (End)

Robert Israel, Table of n, a(n) for n = 1..719 (n=1..359 from Antti Karttunen)

Cf. A107067, A067824.

f:= proc(n) local t, C, x, S;
  C:= map(m -> numtheory:-cyclotomic(m, x), numtheory:-divisors(n) );
  t:= 0:
  S:= combinat:-subsets(C);
  while not S[finished] do
  if map(abs,{coeffs(expand(convert(S[nextvalue](), `*`)), x)}) = {1} then
    t:= t+1;
  fi
od;
t
end proc:
map(f, [$1..100]); # Robert Israel, Aug 24 2017

for(n=1, 359, m=0; p=x^n-1; nE=numdiv(n); P=factor(p); E=P[, 2]; P=P[, 1]; forvec(v=vector(nE, i, [0, E[i]]), divp=prod(k=1, nE, P[k]^v[k]); m++; for(j=0, poldegree(divp), divpcof=polcoeff(divp, j); if(divpcof<-1 || divpcof>1, m--; break))); write("b107748.txt", n, " ", m)); \\ Antti Karttunen, Aug 24 2017, after Herman Jamke's PARI-program for A107067

cp's OEIS Frontend

A378223 Inverse Möbius transform of A345182.

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A124433 Irregular array {a(n,m)} read by rows where (sum{n>=1} sum{m=1 to A001222(n)+1} a(n,m)*y^m/n^x) = 1/(zeta(x)-1+1/y) for all x and y where the double sum converges.

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A317100 Number of series-reduced planted achiral trees with n leaves spanning an initial interval of positive integers.

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PARI

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A342494 Number of compositions of n with strictly decreasing first quotients.

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A342526 Heinz numbers of integer partitions with weakly decreasing first quotients.

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A343371 a(n) = 1 + Sum_{d|n, d < n} a(d - 1).

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Maple

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A343661 Sum of numbers of y-multisets of divisors of x for each x >= 1, y >= 0, x + y = n.

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A343939 Number of n-chains of divisors of n.

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