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A387177 Numbers whose prime indices have choosable sets of strict integer partitions. Positions of nonzero terms in A387115.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98

Offset: 1

Gus Wiseman, Aug 29 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 prime indices of 50 are {1,3,3}, and {(1),(3),(2,1)} is a valid choice of distinct strict partitions, so 50 is in the sequence.

The version for all partitions appears to be A276078, counted by A052335.
The complement for all partitions appears to be A276079, counted by A387134.
The complement for divisors is A355740, counted by A370320.
Twice-partitions of this type (into distinct strict partitions) are counted by A358914.
The version for divisors is A368110, counted by A239312.
The version for initial intervals is A387112, counted by A238873, see A387111.
The complement for initial intervals is A387113, counted by A387118.
These are the positions of nonzero terms in A387115.
The complement is A387176.
Partitions of this type are counted by A387178, complement A387137.
The complement for constant partitions is A387180, counted by A387329, see A387120.
The version 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.
A289509 lists numbers with relatively prime prime indices.
Cf. A000720, A120383, A270995, A299200, A335433, A335448, A357978, A357982, A383706, A387110.

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

A387178 Number of integer partitions of n whose parts have choosable sets of strict integer partitions.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 8, 10, 13, 17, 21, 27, 34, 42, 53, 65, 80, 98, 119, 146, 177, 213, 258, 309, 370, 443, 528, 628, 745, 882, 1043, 1229, 1447, 1700, 1993, 2333, 2727, 3182, 3707, 4311, 5008, 5808, 6727, 7782, 8990, 10371, 11952, 13756, 15815, 18161

Offset: 0

Gus Wiseman, Sep 02 2025

First differs from A052337 in having 745 instead of 746.
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.
a(n) is the number of integer partitions of n such that it is possible to choose a sequence of distinct strict integer partitions of each part.
Also the number of integer partitions of n with no part k whose multiplicity exceeds A000009(k).

			The partition y = (3,3,2) has sets of strict integer partitions ({(2,1),(3)},{(2,1),(3)},{(2)}), and we have the choice ((2,1),(3),(2)) or ((3),(2,1),(2)), so y is counted under a(8).
The a(1) = 1 through a(9) = 10 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (3,3)    (4,3)    (4,4)    (5,4)
                          (4,1)  (4,2)    (5,2)    (5,3)    (6,3)
                                 (5,1)    (6,1)    (6,2)    (7,2)
                                 (3,2,1)  (3,3,1)  (7,1)    (8,1)
                                          (4,2,1)  (3,3,2)  (4,3,2)
                                                   (4,3,1)  (4,4,1)
                                                   (5,2,1)  (5,3,1)
                                                            (6,2,1)
                                                            (3,3,2,1)

For initial intervals instead of strict partitions we have A238873, ranks A387112.
For divisors instead of strict partitions we have A239312, ranks A368110.
The complement for divisors is A370320, ranks A355740.
For prime factors instead of strict partitions we have A370592, ranks A368100.
The complement for prime factors is A370593, ranks A355529.
The complement for initial intervals is A387118, ranks A387113.
The complement for all partitions is A387134, ranks A387577.
The complement is counted by A387137, ranks A387176.
These partitions are ranked by A387177.
For all partitions instead of just strict partitions we have A387328, ranks A387576.
The complement for constant partitions is A387329, ranks A387180.
For constant partitions instead of strict partitions we have A387330, ranks A387181.
A000041 counts integer partitions, strict A000009.
A358914 counts twice-partitions into distinct strict partitions.
A367902 counts choosable set-systems, complement A367903.
Cf. A005703, A052335, A261049, A270995, A276078, A335448, A367867, A367901, A387115.

strptns[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Select[Tuples[strptns/@#],UnsameQ@@#&]!={}&]],{n,0,15}]

A387180 Numbers of which it is not possible to choose a different constant integer partition of each prime index.

Original entry on oeis.org

4, 8, 12, 16, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 125, 128, 132, 135, 136, 140, 144, 148, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 200, 204

Offset: 1

Gus Wiseman, Aug 30 2025

First differs from A276079 in having 125.
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.
Also numbers n with at least one prime index k such that the multiplicity of prime(k) in the prime factorization of n exceeds the number of divisors of k.

			The prime indices of 60 are {1,1,2,3}, and we have the following 4 choices of constant partitions:
  ((1),(1),(2),(3))
  ((1),(1),(2),(1,1,1))
  ((1),(1),(1,1),(3))
  ((1),(1),(1,1),(1,1,1))
Since none of these is strict, 60 is in the sequence.
The prime indices of 90 are {1,2,2,3}, and we have the following 4 strict choices:
  ((1),(2),(1,1),(3))
  ((1),(2),(1,1),(1,1,1))
  ((1),(1,1),(2),(3))
  ((1),(1,1),(2),(1,1,1))
So 90 is not in the sequence.

For prime factors instead of constant partitions we have A355529, counted by A370593.
For divisors instead of constant partitions we have A355740, counted by A370320.
The complement for prime factors is A368100, counted by A370592.
The complement for divisors is A368110, counted by A239312.
The complement for initial intervals is A387112, counted by A238873.
For initial intervals instead of partitions we have A387113, counted by A387118.
These are the positions of zero in A387120.
For strict instead of constant partitions we have A387176, counted by A387137.
The complement for strict partitions is A387177, counted by A387178.
Twice-partitions of this type are counted by A387179, constant-block case of A296122.
The complement is A387181 (nonzeros of A387120), counted by A387330.
Partitions of this type are counted by A387329.
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. A355739, A367771, A387111, A387115.
Cf. A000005, A052335, A063834, A276079, A299200, A299201, A335433, A335448, A355731, A383706, A387110.

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

A387244 Expansion of e.g.f. exp(x^2/(1-x)^4).

Original entry on oeis.org

1, 0, 2, 24, 252, 2880, 38280, 594720, 10565520, 209502720, 4558407840, 107702179200, 2744400415680, 75016089308160, 2189152249764480, 67906418407027200, 2230160988344889600, 77271779968704921600, 2815893910009609228800, 107629691727791474841600, 4304364116456244429388800

Offset: 0

Vaclav Kotesovec, Aug 24 2025

In general, if s >= 1, 1 <= r <= s and e.g.f. = exp(x^r/(1-x)^s) then for n > 0, a(n) = n! * Sum_{k=1..n} binomial(n + (s-r)*k - 1, s*k - 1)/k!.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A293012, A000262, A082579, A091695, A361283.

Cf. A052845, A052887, A386514.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x^2/(1-x)^4))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Aug 25 2025

nmax=20; CoefficientList[Series[E^(x^2/(1-x)^4), {x, 0, nmax}], x] * Range[0, nmax]!
nmax=20; Join[{1}, Table[n!*Sum[Binomial[n+2*k-1, 4*k-1]/k!, {k, 1, n}], {n, 1, nmax}]]
Join[{1}, Table[n!*n*(n - 1)*(n + 1)/6 * HypergeometricPFQ[{1 - n/2, 3/2 - n/2, 1 + n/2, 3/2 + n/2}, {5/4, 3/2, 7/4, 2}, 1/16], {n, 1, 20}]]

For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n + 2*k - 1, 4*k - 1)/k!.
a(n) = 5*(n-1)*a(n-1) - 2*(n-1)*(5*n-11)*a(n-2) + 2*(n-2)*(n-1)*(5*n-14)*a(n-3) - 5*(n-4)*(n-3)*(n-2)*(n-1)*a(n-4) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ 2^(1/5) * 5^(-1/2) * exp(1/80 - 2^(-9/5)*n^(2/5)/3 + 5*2^(-8/5)*n^(4/5) - n) * n^(n - 1/10).

A387289 Decimal expansion of Sum_{n>=1} (-1)^(n+1) P(3n)/(3n), where P(x) is the prime zeta function.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 25 2025

			0.055613262596277710178747463453...

Cf. A387293.

RealDigits[Log[Zeta[3]/Zeta[6]]/3, 10, 105, -1][[1]]

Equals log(zeta(3)/zeta(6))/3.
Equals log(3*(35*zeta(3))^(1/3)/Pi^2).
Sum_{p prime} Sum_{n>=1} (-1)^(n+1)/p^(3*n)/(3*n) = Sum_{p prime} log((1+1/p^3))/3 = log(Product_{p prime} (1+1/p^3))/3 = log(zeta(3)/zeta(6))/3. - Amiram Eldar, Aug 25 2025

A387293 Decimal expansion of Sum_{k>=2} P(k)/zeta(k), where P(k) is the prime zeta function.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 25 2025

			0.5589554346445038949890731220538433207896034...

Cf. A387289.

A387307 a(n) = Sum_{k=0..n} 2^k * binomial(n+2,k+2) * binomial(2*k+4,k+4).

Original entry on oeis.org

1, 15, 166, 1650, 15615, 144025, 1309084, 11793780, 105663885, 943298675, 8401596258, 74716379270, 663813240363, 5894026429725, 52314876771960, 464261939106600, 4119843554861913, 36560929542771735, 324489293583792990, 2880380080564191450, 25572856871556696471

Offset: 0

Seiichi Manyama, Aug 25 2025

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

Cf. A331793, A387308.

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

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

a(n) = sum(k=0, n, 2^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) - 9*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 4^k * 5^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+5*x+4*x^2)^(n+2).
E.g.f.: exp(5*x) * BesselI(2, 4*x) / 4, with offset 2.

A387308 a(n) = Sum_{k=0..n} 2^k * binomial(n+3,k+3) * binomial(2*k+6,k+6).

Original entry on oeis.org

1, 20, 270, 3100, 32711, 328440, 3195276, 30433800, 285604605, 2651696300, 24425110138, 223636254660, 2038173128355, 18508562948400, 167596683043032, 1514149108652880, 13654464563944377, 122951462526317700, 1105768912324277670, 9934852156019798700, 89186900539764803391

Offset: 0

Seiichi Manyama, Aug 25 2025

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

Cf. A331793, A387307.

[&+[2^k * Binomial(n+3,k+3) * Binomial(2*k+6,k+6): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 30 2025

Table[Sum[2^k*Binomial[n+3,k+3]*Binomial[2*k+6,k+6],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 30 2025 *)

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

n*(n+6)*a(n) = (n+3) * (5*(2*n+5)*a(n-1) - 9*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 4^k * 5^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+5*x+4*x^2)^(n+3).
E.g.f.: exp(5*x) * BesselI(3, 4*x) / 8, with offset 3.

A387309 a(n) = Sum_{k=0..n} 3^k * binomial(n+1,k+1) * binomial(2*k+2,k+2).

Original entry on oeis.org

1, 14, 174, 2128, 26045, 320082, 3951493, 48987848, 609592347, 7610525650, 95287524332, 1196054790168, 15046318739803, 189654839753750, 2394743468261190, 30285593026553536, 383554551776056139, 4863775493104574634, 61748210178809072722, 784757334938247965840, 9983152795673915802399

Offset: 0

Seiichi Manyama, Aug 25 2025

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

Cf. A387310, A387311.
Cf. A026376, A331793.
Cf. A385716.

[&+[3^k * Binomial(n+1,k+1) * Binomial(2*k+2,k+2): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 30 2025

Table[Sum[3^k*Binomial[n+1,k+1]*Binomial[2*k+2,k+2],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 30 2025 *)

a(n) = sum(k=0, n, 3^k*binomial(n+1, k+1)*binomial(2*k+2, k+2));

G.f.: ((1-7*x)/sqrt((1-x) * (1-13*x)) - 1)/(18*x^2).
n*(n+2)*a(n) = (n+1) * (7*(2*n+1)*a(n-1) - 13*n*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = [x^n] (1+7*x+9*x^2)^(n+1).
E.g.f.: exp(7*x) * BesselI(1, 6*x) / 3, with offset 1.

A387313 Expansion of 1/((1-x) * (1-9*x))^(5/2).

Original entry on oeis.org

1, 25, 415, 5775, 72870, 864150, 9818130, 108109650, 1162302735, 12262882775, 127424209913, 1307536637225, 13276264807260, 133597932407100, 1334029357684980, 13231465264538100, 130461712570627245, 1279632533997010725, 12492837802976030115, 121456026730456739475

Offset: 0

Seiichi Manyama, Aug 25 2025

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

Cf. A084771, A331516, A387314.

Cf. A387229, A387307.

R := PowerSeriesRing(Rationals(), 34); f := 1/((1-x) * (1-9*x))^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025

CoefficientList[Series[1/((1-x)*(1-9*x))^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)

my(N=20, x='x+O('x^N)); Vec(1/((1-x)*(1-9*x))^(5/2))

n*a(n) = (10*n+15)*a(n-1) - 9*(n+3)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-5/2,k) * binomial(-5/2,n-k).
a(n) = Sum_{k=0..n} (-8)^k * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = Sum_{k=0..n} 8^k * 9^(n-k) * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = (binomial(n+4,2)/6) * A387307(n).
a(n) = (-1)^n * Sum_{k=0..n} 10^k * (9/10)^(n-k) * binomial(-5/2,k) * binomial(k,n-k).

cp's OEIS Frontend

A387177 Numbers whose prime indices have choosable sets of strict integer partitions. Positions of nonzero terms in A387115.

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A387178 Number of integer partitions of n whose parts have choosable sets of strict integer partitions.

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A387180 Numbers of which it is not possible to choose a different constant integer partition of each prime index.

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A387244 Expansion of e.g.f. exp(x^2/(1-x)^4).

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A387289 Decimal expansion of Sum_{n>=1} (-1)^(n+1) P(3*n)/(3*n), where P(x) is the prime zeta function.

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A387293 Decimal expansion of Sum_{k>=2} P(k)/zeta(k), where P(k) is the prime zeta function.

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A387307 a(n) = Sum_{k=0..n} 2^k * binomial(n+2,k+2) * binomial(2*k+4,k+4).

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A387308 a(n) = Sum_{k=0..n} 2^k * binomial(n+3,k+3) * binomial(2*k+6,k+6).

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A387309 a(n) = Sum_{k=0..n} 3^k * binomial(n+1,k+1) * binomial(2*k+2,k+2).

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A387289 Decimal expansion of Sum_{n>=1} (-1)^(n+1) P(3n)/(3n), where P(x) is the prime zeta function.