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A387112 Numbers with (strictly) choosable initial intervals of prime indices.

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95

Offset: 1

Gus Wiseman, Aug 23 2025

First differs from A371088 in having a(86) = 121.
The initial interval of a nonnegative integer x is the set {1,...,x}.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We say that a set or sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1,2,3},{1},{1,3},{2}) is not.
This sequence lists all numbers k such that if the prime indices of k are (x1,x2,...,xz), then the sequence of sets (initial intervals) ({1,...,x1},{1,...,x2},...,{1,...,xz}) is choosable.

			The prime indices of 85 are {3,7}, with initial intervals {{1,2,3},{1,2,3,4,5,6,7}}, which are choosable, so 85 is in the sequence
The prime indices of 90 are {1,2,2,3}, with initial intervals {{1},{1,2},{1,2},{1,2,3}}, which are not choosable, so 90 is not in the sequence.

Partitions of this type are counted by A238873, complement A387118.
For partitions instead of initial intervals we have A276078, complement A276079.
For prime factors instead of initial intervals we have A368100, complement A355529.
For divisors instead of initial intervals we have A368110, complement A355740.
These are all the positions of nonzero terms in A387111, complement A387134.
The complement is A387113.
For strict partitions instead of initial intervals we have A387176, complement A387137.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A367902 counts choosable set-systems, complement A367903.
A370582 counts sets with choosable prime factors, complement A370583.
A370585 counts maximal subsets with choosable prime factors.
Cf. A003963, A005703, A052335, A239312, A335433, A335448, A355739, A387110, A383706.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[Range/@prix[#]],UnsameQ@@#&]!={}&]

A387133 Number of ways to choose a sequence of distinct integer partitions, one of each prime factor of n (with multiplicity).

Original entry on oeis.org

1, 2, 3, 2, 7, 6, 15, 0, 6, 14, 56, 6, 101, 30, 21, 0, 297, 12, 490, 14, 45, 112, 1255, 0, 42, 202, 6, 30, 4565, 42, 6842, 0, 168, 594, 105, 12, 21637, 980, 303, 0, 44583, 90, 63261, 112, 42, 2510, 124754, 0, 210, 84, 891, 202, 329931, 12, 392, 0, 1470, 9130

Offset: 1

Gus Wiseman, Aug 26 2025

			The prime factors of 9 are (3,3), and the a(9) = 6 choices are:
  ((3),(2,1))
  ((3),(1,1,1))
  ((2,1),(3))
  ((2,1),(1,1,1))
  ((1,1,1),(3))
  ((1,1,1),(2,1))

For prime factors instead of partitions we have A008966, see A355741.
Twice partitions of this type are counted by A296122.
For prime indices instead of factors we have A387110, see A387136.
For strict partitions and prime indices we have A387115.
For constant partitions and prime indices we have A387120.
Positions of zero are A387326, for indices apparently A276079 (complement A276078).
Allowing repeated partitions gives A387327, see A299200, A357977.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A063834, A261049, A299201, A335433, A335448, A355739, A383706, A387111, A387135.

Table[Length[Select[Tuples[IntegerPartitions/@Flatten[ConstantArray@@@FactorInteger[n]]],UnsameQ@@#&]],{n,30}]

A387134 Number of integer partitions of n whose parts do not have choosable sets of integer partitions.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 6, 8, 12, 17, 25, 34, 49, 65, 89, 118, 158, 206, 271, 349, 453, 578, 740, 935, 1186, 1486, 1865, 2322, 2890, 3572, 4415, 5423, 6659, 8134, 9927, 12062, 14643, 17706, 21387, 25746, 30957, 37109, 44433, 53054, 63273, 75276, 89444, 106044

Offset: 0

Gus Wiseman, Aug 29 2025

Number of integer partitions of n such that it is not possible to choose a sequence of distinct integer partitions, one of each part.

Also the number of integer partitions of n with at least one part k satisfying that the multiplicity of k exceeds the number of integer partitions of k.

			The a(2) = 1 through a(8) = 12 partitions:
  (11)  (111)  (211)   (311)    (222)     (511)      (611)
               (1111)  (2111)   (411)     (2221)     (2222)
                       (11111)  (2211)    (3211)     (3311)
                                (3111)    (4111)     (4211)
                                (21111)   (22111)    (5111)
                                (111111)  (31111)    (22211)
                                          (211111)   (32111)
                                          (1111111)  (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

These partitions are ranked by A276079.
For divisors instead of partitions we have A370320, complement A239312.
The complement for prime factors is A370592, ranks A368100.
For prime factors instead of partitions we have A370593, ranks A355529.
For initial intervals instead of partitions we have A387118, complement A238873.
For just choices of strict partitions we have A387137.
The complement is counted by A387328, ranks A276078.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
Cf. A335433, A355535, A367867, A367901, A367903, A367905, A367907, A370583, A370594.

Table[Length[Select[IntegerPartitions[n],Length[Select[Tuples[IntegerPartitions/@#],UnsameQ@@#&]]==0&]],{n,0,15}]

A387176 Numbers whose prime indices do not have choosable sets of strict integer partitions. Zeros of A387115.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 45, 48, 52, 54, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 162, 164, 168, 171, 172

Offset: 1

Gus Wiseman, Aug 27 2025

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.

The complement for all partitions appears to be A276078, counted by A052335.
For all partitions we appear to have A276079, counted by A387134.
For divisors instead of strict partitions we have A355740, counted by A370320.
Twice-partitions of this type (into distinct strict partitions) are counted by A358914.
The complement for divisors is A368110, counted by A239312.
The complement for initial intervals is A387112, counted by A238873, see A387111.
For initial intervals instead of strict partitions we have A387113, counted by A387118.
These are the positions of 0 in A387115.
Partitions of this type are counted by A387137, complement A387178.
The complement is A387177.
The version for constant partitions is A387180, counted by A387329.
The complement for constant partitions is A387181, counted by A387330.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together the prime indices of n.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A000720, A261049, A270995, A335433, A335448, A355744, A357978, A357980, A383706, A387110, A387120.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@prix[#]],UnsameQ@@#&]=={}&]

A387228 Expansion of sqrt((1-x) / (1-5*x)^5).

Original entry on oeis.org

1, 12, 103, 764, 5215, 33728, 210021, 1271504, 7532547, 43859460, 251809701, 1428911652, 8028877233, 44734340784, 247433518875, 1359902816880, 7432212863235, 40416897046740, 218812616979845, 1179889937796900, 6339243523221245, 33947223885549040, 181245459484155935

Offset: 0

Seiichi Manyama, Aug 23 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A387229, A387230.
Cf. A085362, A163869.
Cf. A377199.

R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- x) / (1-5*x)^5); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 24 2025

CoefficientList[Series[Sqrt[(1-x)/(1-5*x)^5],{x,0,33}],x] (* Vincenzo Librandi, Aug 24 2025 *)

my(N=30, x='x+O('x^N)); Vec(sqrt((1-x)/(1-5*x)^5))

n*a(n) = (6*n+6)*a(n-1) - 5*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n+1,n-k).
a(n) ~ 8 * 5^(n - 1/2) * n^(3/2) / (3*sqrt(Pi)). - Vaclav Kotesovec, Aug 23 2025

A387273 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n+3,k+3) * binomial(2*k+6,k+6).

Original entry on oeis.org

1, 16, 165, 1400, 10661, 75936, 517524, 3420960, 22123530, 140782048, 885008839, 5511579528, 34073731965, 209428887360, 1281220578936, 7808422173120, 47440778110398, 287490594872160, 1738463164498410, 10493677382085744, 63245915436539682, 380697445274657984

Offset: 0

Seiichi Manyama, Aug 24 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A344055, A387272, A387274.

Cf. A387239, A387276.

[&+[2^(n-k) * Binomial(n+3,k+3) * Binomial(2*k+6,k+6): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025

Table[Sum[2^(n-k)*Binomial[n+3,k+3]*Binomial[2*k+6,k+6],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)

a(n) = sum(k=0, n, 2^(n-k)*binomial(n+3, k+3)*binomial(2*k+6, k+6));

n*(n+6)*a(n) = (n+3) * (4*(2*n+5)*a(n-1) - 12*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+4*x+x^2)^(n+3).
E.g.f.: exp(4*x) * BesselI(3, 2*x), with offset 3.

A387274 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n+4,k+4) * binomial(2*k+8,k+8).

Original entry on oeis.org

1, 20, 246, 2408, 20636, 162288, 1203000, 8546208, 58823919, 395245708, 2606333730, 16933021560, 108703640136, 691068080928, 4358220121296, 27301946599872, 170074452183570, 1054434358722024, 6510869338671852, 40063301434583504, 245781459952640040

Offset: 0

Seiichi Manyama, Aug 24 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A344055, A387272, A387273.

[&+[2^(n-k) * Binomial(n+4,k+4) * Binomial(2*k+8,k+8): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025

Table[Sum[2^(n-k)*Binomial[n+4,k+4]*Binomial[2*k+8,k+8],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)

a(n) = sum(k=0, n, 2^(n-k)*binomial(n+4, k+4)*binomial(2*k+8, k+8));

n*(n+8)*a(n) = (n+4) * (4*(2*n+7)*a(n-1) - 12*(n+3)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n+4,n-2*k) * binomial(2*k+4,k).
a(n) = [x^n] (1+4*x+x^2)^(n+4).
E.g.f.: exp(4*x) * BesselI(4, 2*x), with offset 4.

A387275 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+2,k+2) * binomial(2*k+4,k+4).

Original entry on oeis.org

1, 15, 154, 1350, 10890, 83650, 623056, 4547520, 32735085, 233369675, 1652203542, 11638252730, 81674873553, 571575363975, 3991529920440, 27829484027400, 193791573179883, 1348196149698885, 9372495529924710, 65120144658997050, 452263192928596896

Offset: 0

Seiichi Manyama, Aug 24 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A387276, A387277, A387278.

[&+[3^(n-k) * Binomial(n+2,k+2) * Binomial(2*k+4,k+4): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025

Table[Sum[3^(n-k)*Binomial[n+2,k+2]*Binomial[2*k+4,k+4],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)

a(n) = sum(k=0, n, 3^(n-k)*binomial(n+2, k+2)*binomial(2*k+4, k+4));

n*(n+4)*a(n) = (n+2) * (5*(2*n+3)*a(n-1) - 21*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+5*x+x^2)^(n+2).
E.g.f.: exp(5*x) * BesselI(2, 2*x), with offset 2.
a(n) ~ 7^(n + 5/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 31 2025

A387277 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+4,k+4) * binomial(2*k+8,k+8).

Original entry on oeis.org

1, 25, 381, 4585, 47978, 458010, 4100370, 35027850, 288845370, 2317794050, 18203687502, 140533725150, 1069904389008, 8052575725680, 60033791987424, 444015014417280, 3261950250436845, 23827019766988725, 173193081555808545, 1253583401573658925, 9040278899072328006

Offset: 0

Seiichi Manyama, Aug 24 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A387275, A387276, A387278.

Cf. A387274.

[&+[3^(n-k) * Binomial(n+4,k+4) * Binomial(2*k+8,k+8): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 30 2025

Table[Sum[3^(n-k)*Binomial[n+4,k+4]*Binomial[2*k+8,k+8],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 30 2025 *)

a(n) = sum(k=0, n, 3^(n-k)*binomial(n+4, k+4)*binomial(2*k+8, k+8));

n*(n+8)*a(n) = (n+4) * (5*(2*n+7)*a(n-1) - 21*(n+3)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+4,n-2*k) * binomial(2*k+4,k).
a(n) = [x^n] (1+5*x+x^2)^(n+4).
E.g.f.: exp(5*x) * BesselI(4, 2*x), with offset 4.

A387310 a(n) = Sum_{k=0..n} 3^k * binomial(n+2,k+2) * binomial(2*k+4,k+4).

Original entry on oeis.org

1, 21, 330, 4690, 63690, 844662, 11052496, 143462592, 1852852365, 23853938185, 306473670822, 3932435239278, 50417223635233, 646085510253645, 8277409340709240, 106037993391958936, 1358437551566242347, 17404555385537336583, 223025734596708637750, 2858460480570547144110

Offset: 0

Seiichi Manyama, Aug 25 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A387309, A387311.

[&+[3^k * Binomial(n+2,k+2) * Binomial(2*k+4,k+4): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025

Table[Sum[3^k * Binomial[n+2,k+2]*Binomial[2*k+4, k+4],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *)

a(n) = sum(k=0, n, 3^k*binomial(n+2, k+2)*binomial(2*k+4, k+4));

n*(n+4)*a(n) = (n+2) * (7*(2*n+3)*a(n-1) - 13*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+7*x+9*x^2)^(n+2).
E.g.f.: exp(7*x) * BesselI(2, 6*x) / 9, with offset 2.

cp's OEIS Frontend

A387112 Numbers with (strictly) choosable initial intervals of prime indices.

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A387133 Number of ways to choose a sequence of distinct integer partitions, one of each prime factor of n (with multiplicity).

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A387134 Number of integer partitions of n whose parts do not have choosable sets of integer partitions.

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A387176 Numbers whose prime indices do not have choosable sets of strict integer partitions. Zeros of A387115.

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A387228 Expansion of sqrt((1-x) / (1-5*x)^5).

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A387273 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n+3,k+3) * binomial(2*k+6,k+6).

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A387274 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n+4,k+4) * binomial(2*k+8,k+8).

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A387275 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+2,k+2) * binomial(2*k+4,k+4).

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A387277 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+4,k+4) * binomial(2*k+8,k+8).