A364891 -id:A364891 - cp's OEIS frontend

A364893 a(n) is the minimal positive value of m such that A325433(2m, 2n+1) > A364891(2m, 2n+1).

11, 28, 54, 88, 129, 179, 237, 303, 376, 458, 548, 646, 752, 866, 988, 1118, 1256, 1402, 1558, 1719, 1889, 2067, 2253, 2447, 2650, 2860, 3078, 3304, 3539, 3781, 4031, 4289, 4556, 4830, 5112, 5403, 5701, 6007, 6332, 6644, 6975, 7313, 7659, 8014, 8376, 8747, 9125

Offset: 1

Stefano Spezia, Aug 12 2023

nonn

K. Banerjee and M. G. Dastidar, Inequalities for the partition function arising from truncated theta series, RISC Report Series No. 22-20, 2023.

Cf. A000041, A325433, A364891, A364894.

A325433[n_, k_]:=(-1)^(k-1)*Sum[(-1)^j*(PartitionsP[n-j*(3*j+1)/2]-PartitionsP[n-j*(3*j+5)/2-1]), {j, 0, k-1}];
A364891[n_, k_]:=(-1)^(k-1)*Sum[(-1)^j*(PartitionsP[n-j(2j+1)]-PartitionsP[n-(j+1)(2j+1)]), {j, 0, k-1}];
nmax=47; a={}; For[n=1,n<=nmax,n++,m=1;While[A325433[2m,2n+1]<=A364891[2m,2n+1],m++]; AppendTo[a,m]]; a

Empirical: a(n) ~ A364894(n). (See p. 5 in Banerjee and Dastidar.)

A364892 Row sums of A364891.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 4, 5, 8, 10, 15, 19, 27, 34, 47, 59, 79, 99, 130, 162, 210, 260, 332, 410, 517, 635, 794, 970, 1202, 1463, 1799, 2180, 2664, 3214, 3904, 4693, 5669, 6789, 8163, 9740, 11658, 13865, 16527, 19592, 23267, 27496, 32538, 38343, 45223, 53142, 62488

Offset: 1

Stefano Spezia, Aug 12 2023

nonn

Cf. A000041, A325434, A364891.

A364891[n_, k_]:=(-1)^(k-1)*Sum[(-1)^j*(PartitionsP[n-j(2j+1)]-PartitionsP[n-(j+1)(2j+1)]), {j, 0, k-1}]; Table[Sum[A364891[n,k],{k,1,n}],{n,1,51}]

a(n) = Sum_{k=1..n} ((-1)^(k-1)*Sum_{j=0..k-1} (-1)^j*(p(n - j*(2*j + 1)) - p(n - (j + 1)*(2*j + 1)))), where p(n) = A000041(n) is the number of partitions of n.

Conjecture: lim_{n->oo} a(n)/A000041(n) = 1/4.

A364894 a(n) = floor(4n^2 + 7n - sqrt(n)*log(n)) - floor(n/3).

Original entry on oeis.org

11, 29, 54, 88, 130, 179, 237, 304, 377, 459, 550, 647, 753, 868, 989, 1119, 1258, 1403, 1558, 1720, 1890, 2068, 2254, 2448, 2650, 2861, 3078, 3305, 3539, 3781, 4031, 4290, 4555, 4830, 5112, 5402, 5701, 6007, 6321, 6643, 6974, 7311, 7658, 8012, 8374, 8745, 9123

Offset: 1

Stefano Spezia, Aug 12 2023

nonn

K. Banerjee and M. G. Dastidar, Inequalities for the partition function arising from truncated theta series, RISC Report Series No. 22-20, 2023. See p. 5, 2nd table.

Cf. A002264, A325433, A364891, A364893.

Table[Floor[4n^2+7n-Sqrt[n]Log[n]]-Floor[n/3],{n,47}]

cp's OEIS Frontend

A364893 a(n) is the minimal positive value of m such that A325433(2m, 2n+1) > A364891(2m, 2n+1).

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A364892 Row sums of A364891.

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A364894 a(n) = floor(4n^2 + 7n - sqrt(n)*log(n)) - floor(n/3).

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cp's OEIS Frontend

A364893 a(n) is the minimal positive value of m such that A325433(2m, 2n+1) > A364891(2m, 2n+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A364892 Row sums of A364891.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A364894 a(n) = floor(4*n^2 + 7*n - sqrt(n)*log(n)) - floor(n/3).

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A364894 a(n) = floor(4n^2 + 7n - sqrt(n)*log(n)) - floor(n/3).