A052002 -id:A052002 - cp's OEIS frontend

A306956 Sum over all partitions of n into distinct parts of the LCM of the parts.

1, 1, 2, 5, 7, 15, 21, 39, 58, 90, 142, 218, 325, 465, 695, 948, 1411, 1977, 2883, 3940, 5415, 7422, 10126, 14091, 18947, 25666, 34282, 45890, 60710, 82211, 108510, 142960, 185271, 240595, 315158, 409231, 531967, 688689, 880997, 1126451, 1447754, 1849743

Offset: 0

Alois P. Heinz, Mar 17 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Partition (number theory)

Cf. A000009, A000041, A001560, A040051, A052002, A181844, A319301.

b:= proc(n, i, r) option remember; `if`(i*(i+1)/2 b(n$2, 1):
seq(a(n), n=0..44);

b[n_, i_, r_] := b[n, i, r] = If[i(i+1)/2 < n, 0, If[n == 0, r, b[n, i-1, r] + b[n-i, Min[i-1, n-i], LCM[i, r]]]];
a[n_] := b[n, n, 1];
Table[a[n], {n, 0, 44}] (* Jean-François Alcover, Mar 20 2019, translated from Maple *)

a(n) mod 2 = A040051(n).
a(n) is even <=> n in { A001560 }.
a(n) is odd  <=> n in { A052002 }.

A331230 Numbers k such that the number of factorizations of k into distinct factors > 1 is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 17, 18, 19, 20, 23, 24, 25, 28, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 83, 84, 88, 89, 90, 92, 97, 98, 99, 100, 101, 102

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A319237 in lacking 300.

The version for strict integer partitions is A001318.
The version for integer partitions is A052002.
The version for set partitions appears to be A032766.
The non-strict version is A331050.
The version for primes (instead of odds) is A331201.
The even version is A331231.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with n strict factorizations is A330974(n).
Cf. A035359, A045780, A318286, A328966, A330991, A330997, A331051, A331200, A331232.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],OddQ[Length[strfacs[#]]]&]

A356508 G.f. A(x) satisfies: 2 = Product_{n>=1} (1 + x^nA(x)) (1 + x^(n-1)/A(x)).

Original entry on oeis.org

1, 4, 14, 84, 444, 2928, 18214, 125428, 844534, 5989816, 42186878, 305757288, 2215509018, 16326672796, 120612763510, 900561207232, 6746557569136, 50906726784700, 385432963013140, 2933390906035044, 22395805754363208, 171660252748284852, 1319474586701337644

Offset: 0

Paul D. Hanna, Aug 11 2022

nonn

Conjecture: a(n) == 2 (mod 4) at n = 2*k for all k > 0 that have an odd number of partitions (A052002), otherwise a(n) == 0 (mod 4) when n > 0.

			G.f.: A(x) = 1 + 4*x + 14*x^2 + 84*x^3 + 444*x^4 + 2928*x^5 + 18214*x^6 + 125428*x^7 + 844534*x^8 + 5989816*x^9 + 42186878*x^10 + ...
such that
2 = (1 + x*A(x))*(1 + 1/A(x)) * (1 + x^2*A(x))*(1 + x/A(x)) * (1 + x^3*A(x))*(1 + x^2/A(x)) * (1 + x^4*A(x))*(1 + x^3/A(x)) * (1 + x^5*A(x))*(1 + x^4/A(x)) * ...
also,
2/P(x) = ... + x^10/A(x)^5 + x^6/A(x)^4 + x^3/A(x)^3 + x/A(x)^2 + 1/A(x) + 1 + x*A(x) + x^3*A(x)^2 + x^6*A(x)^3 + x^10*A(x)^4 + ... + x^(n*(n+1)/2) * A(x)^n + ...
where P(x) is the partition function and begins
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...
and
2/P(x) = 2 - 2*x - 2*x^2 + 2*x^5 + 2*x^7 - 2*x^12 - 2*x^15 + 2*x^22 + 2*x^26 - 2*x^35 - 2*x^40 + 2*x^51 + 2*x^57 - 2*x^70 - 2*x^77 + 2*x^92 + 2*x^100 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A356499, A000041, A052002.

(* Calculation of constants {d,c}: *) {1/r, -s*Log[r]/2 * Sqrt[-(r*(1 + s)*(QPochhammer[-s, r]^2* Derivative[0, 1][QPochhammer][-1/s, r] + 2*(1 + s) * Derivative[0, 1][QPochhammer][-s, r])) / (Pi * QPochhammer[-s, r] * (s* Log[r]^2 + (1 + s)^2*(QPolyGamma[1, Log[-1/s]/Log[r], r] + QPolyGamma[1, Log[-s]/Log[r], r])))]} /. FindRoot[{QPochhammer[-1/s, r]*QPochhammer[-s, r] == 2*(1 + s), (1 + s)*(QPolyGamma[0, Log[-1/s]/Log[r], r] - QPolyGamma[0, Log[-s]/Log[r], r]) == s*Log[r]}, {r, 1/8}, {s, 3}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 30 2023 *)

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( -2 + prod(n=1, #A, (1 + x^n*Ser(A)) * (1 + x^(n-1)/Ser(A)) ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

{a(n) = my(A=[1], M, P=prod(k=1, n, 1-x^k +x*O(x^n))); for(i=1, n, A=concat(A, 0); M = ceil(sqrt(2*n+9));
A[#A] = polcoeff( -2*P + sum(m=-M, M, x^(m*(m+1)/2)*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) satisfies:
(1) 2/P(x) = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * A(x)^n, where P(x) = 1/Product_{n>=1} (1 - x^n) is the partition function (A000041)..
(2) 2 = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), by the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 8.2221649228195625... and c = 1.06682907735826... - Vaclav Kotesovec, Mar 14 2023

A127700 Numbers n such that partition number of n^2 is even.

Original entry on oeis.org

3, 5, 8, 10, 16, 20, 21, 23, 26, 28, 30, 39, 41, 43, 45, 46, 47, 48, 49, 50, 51, 56, 58, 59, 63, 66, 68, 70, 71, 73, 76, 78, 80, 82, 84, 85, 86, 87, 88, 92, 93, 95, 96, 97, 100, 102, 103, 111, 112, 113, 115, 117, 120, 121, 122, 123, 125, 127, 129, 131, 134, 135, 137

Offset: 1

Zak Seidov, Apr 03 2007

nonn

Cf. A000041, A001560, A052001, A052002, A127219.

Select[Range[200],EvenQ[PartitionsP[ #^2]]&]

A000041(n^2) is even.

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A306956 Sum over all partitions of n into distinct parts of the LCM of the parts.

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A331230 Numbers k such that the number of factorizations of k into distinct factors > 1 is odd.

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A356508 G.f. A(x) satisfies: 2 = Product_{n>=1} (1 + x^nA(x)) (1 + x^(n-1)/A(x)).

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A127700 Numbers n such that partition number of n^2 is even.

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cp's OEIS Frontend

A306956 Sum over all partitions of n into distinct parts of the LCM of the parts.

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Author

Keywords

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Programs

Maple

Mathematica

Formula

A331230 Numbers k such that the number of factorizations of k into distinct factors > 1 is odd.

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Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A356508 G.f. A(x) satisfies: 2 = Product_{n>=1} (1 + x^n*A(x)) * (1 + x^(n-1)/A(x)).

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A127700 Numbers n such that partition number of n^2 is even.

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Mathematica

Formula

A356508 G.f. A(x) satisfies: 2 = Product_{n>=1} (1 + x^nA(x)) (1 + x^(n-1)/A(x)).