cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A340719 Number of partitions of n into 8 distinct and relatively prime parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 52, 70, 89, 116, 146, 186, 230, 288, 352, 434, 525, 638, 764, 919, 1090, 1297, 1527, 1801, 2104, 2462, 2857, 3319, 3828, 4417, 5066, 5811, 6630, 7563, 8588, 9747, 11018, 12447, 14012, 15760, 17674, 19798, 22122, 24688, 27493, 30573, 33940, 37616
Offset: 36

Views

Author

Ilya Gutkovskiy, Feb 23 2021

Keywords

Crossrefs

Cf. A023022, A023028, A023033, A078374, A101271, A339672, A341868, A341870, A341912, A341913, A341914.

Programs

  • Mathematica
    nmax = 88; CoefficientList[Series[Sum[MoebiusMu[k] x^(36 k)/Product[1 - x^(j k), {j, 1, 8}], {k, 1, nmax}], {x, 0, nmax}], x] // Drop[#, 36] &

Formula

G.f.: Sum_{k>=1} mu(k)* x^(36*k) / Product_{j=1..8} (1 - x^(j*k)).

A341868 Number of partitions of n into 4 distinct and relatively prime parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 22, 27, 33, 39, 45, 54, 61, 72, 79, 94, 101, 120, 127, 149, 158, 185, 189, 225, 231, 267, 274, 321, 319, 378, 377, 435, 439, 511, 495, 588, 577, 661, 656, 764, 729, 863, 836, 954, 939, 1089, 1022, 1215, 1165, 1323, 1289, 1492, 1392, 1650, 1566, 1776, 1715
Offset: 10

Views

Author

Ilya Gutkovskiy, Feb 23 2021

Keywords

Crossrefs

Cf. A000742, A023022, A023024, A078374, A101271, A339672, A340719, A341870, A341912, A341913, A341914.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Sum[MoebiusMu[k] x^(10 k)/Product[1 - x^(j k), {j, 1, 4}], {k, 1, nmax}], {x, 0, nmax}], x] // Drop[#, 10] &

Formula

G.f.: Sum_{k>=1} mu(k)* x^(10*k) / Product_{j=1..4} (1 - x^(j*k)).

A341870 Number of partitions of n into 6 distinct and relatively prime parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 58, 71, 90, 110, 136, 163, 199, 235, 282, 330, 391, 453, 532, 610, 709, 808, 931, 1052, 1206, 1353, 1540, 1718, 1945, 2158, 2432, 2682, 3009, 3305, 3692, 4035, 4493, 4891, 5427, 5883, 6510, 7033, 7758, 8352, 9192, 9862, 10829, 11584, 12687, 13539
Offset: 21

Views

Author

Ilya Gutkovskiy, Feb 23 2021

Keywords

Crossrefs

Cf. A023022, A023026, A023031, A078374, A101271, A339672, A340719, A341868, A341912, A341913, A341914.

Programs

  • Mathematica
    nmax = 76; CoefficientList[Series[Sum[MoebiusMu[k] x^(21 k)/Product[1 - x^(j k), {j, 1, 6}], {k, 1, nmax}], {x, 0, nmax}], x] // Drop[#, 21] &

Formula

G.f.: Sum_{k>=1} mu(k)* x^(21*k) / Product_{j=1..6} (1 - x^(j*k)).

A341912 Number of partitions of n into 5 distinct and relatively prime parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, 70, 83, 101, 118, 141, 162, 192, 218, 255, 286, 333, 370, 427, 470, 540, 590, 673, 730, 831, 894, 1014, 1085, 1224, 1305, 1469, 1552, 1747, 1841, 2057, 2163, 2418, 2520, 2818, 2933, 3256, 3388, 3765, 3879, 4319, 4452, 4914, 5068
Offset: 15

Views

Author

Ilya Gutkovskiy, Feb 23 2021

Keywords

Crossrefs

Cf. A000743, A023022, A023025, A078374, A101271, A339672, A340719, A341868, A341870, A341913, A341914.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Sum[MoebiusMu[k] x^(15 k)/Product[1 - x^(j k), {j, 1, 5}], {k, 1, nmax}], {x, 0, nmax}], x] // Drop[#, 15] &

Formula

G.f.: Sum_{k>=1} mu(k)* x^(15*k) / Product_{j=1..5} (1 - x^(j*k)).
a(n) <= A001401(n-15). - R. J. Mathar, Feb 28 2021

A341913 Number of partitions of n into 9 distinct and relatively prime parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 54, 73, 94, 123, 157, 201, 252, 318, 393, 488, 598, 732, 887, 1076, 1291, 1549, 1845, 2194, 2592, 3060, 3589, 4206, 4904, 5708, 6615, 7657, 8824, 10156, 11648, 13338, 15224, 17354, 19720, 22380, 25330, 28629, 32277, 36347, 40829, 45812, 51291, 57358
Offset: 45

Views

Author

Ilya Gutkovskiy, Feb 23 2021

Keywords

Crossrefs

Cf. A023022, A023029, A023034, A078374, A101271, A339672, A340719, A341868, A341870, A341912, A341914.

Programs

  • Mathematica
    nmax = 97; CoefficientList[Series[Sum[MoebiusMu[k] x^(45 k)/Product[1 - x^(j k), {j, 1, 9}], {k, 1, nmax}], {x, 0, nmax}], x] // Drop[#, 45] &

Formula

G.f.: Sum_{k>=1} mu(k)* x^(45*k) / Product_{j=1..9} (1 - x^(j*k)).

A300274 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} (1 + x^n)/(1 - x^n).

Original entry on oeis.org

2, 2, 6, 10, 22, 30, 62, 86, 146, 206, 342, 454, 726, 974, 1442, 1962, 2862, 3762, 5398, 7094, 9834, 12942, 17726, 22938, 31042, 40094, 53254, 68518, 90246, 114914, 150142, 190550, 245906, 310942, 398554, 500474, 637590, 797534, 1007714, 1255850, 1578526, 1956786
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 01 2018

Keywords

Comments

Moebius transform of A015128.

Links

Crossrefs

Cf. A000837, A015128, A078374, A300275, A300276, A300277, A300278.

Programs

  • Mathematica
    nn = 42; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Product[(1 + x^n)/(1 - x^n), {n, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
s[n_] := SeriesCoefficient[Product[(1 + x^k)/(1 - x^k), {k, 1, n}], {x, 0, n}]; a[n_] := Sum[MoebiusMu[n/d] s[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 42}]

Formula

a(n) = Sum_{d|n} mu(n/d)*A015128(d).

A300276 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} (1 + x^n)^n.

Original entry on oeis.org

1, 1, 4, 6, 15, 22, 48, 75, 137, 218, 384, 593, 1003, 1549, 2501, 3857, 6110, 9256, 14408, 21675, 33081, 49422, 74483, 110135, 164116, 240955, 355027, 517553, 755893, 1093649, 1584518, 2277986, 3274887, 4679619, 6682635, 9491959, 13471238, 19030370, 26849913, 37734570
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 01 2018

Keywords

Comments

Moebius transform of A026007.

Links

Crossrefs

Cf. A000837, A026007, A078374, A300274, A300275, A300277, A300278.

Programs

  • Mathematica
    nn = 40; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Product[(1 + x^n)^n, {n, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
s[n_] := SeriesCoefficient[Product[(1 + x^k)^k, {k, 1, n}], {x, 0, n}]; a[n_] := Sum[MoebiusMu[n/d] s[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 40}]

Formula

a(n) = Sum_{d|n} mu(n/d)*A026007(d).

A300278 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} (1 + n*x^n).

Original entry on oeis.org

1, 1, 4, 5, 14, 19, 42, 57, 115, 170, 287, 433, 694, 1061, 1709, 2461, 3740, 5635, 8243, 12256, 18255, 26135, 37826, 54209, 78315, 110488, 159418, 224514, 315414, 442790, 618665, 855640, 1199409, 1642334, 2288904, 3144738, 4303994, 5862294, 8031872, 10869290, 14749050
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 01 2018

Keywords

Comments

Moebius transform of A022629.

Links

Crossrefs

Cf. A000837, A022629, A078374, A085410, A300274, A300275, A300276, A300277.

Programs

  • Mathematica
    nn = 41; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Product[(1 + n x^n), {n, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
s[n_] := SeriesCoefficient[Product[(1 + k x^k), {k, 1, n}], {x, 0, n}]; a[n_] := Sum[MoebiusMu[n/d] s[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 41}]

Formula

a(n) = Sum_{d|n} mu(n/d)*A022629(d).

A366852 Number of integer partitions of n into odd parts with a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 4, 0, 1, 4, 1, 2, 6, 1, 1, 6, 3, 1, 8, 2, 1, 13, 1, 0, 13, 1, 7, 15, 1, 1, 19, 6, 1, 25, 1, 2, 33, 1, 1, 32, 5, 10, 39, 2, 1, 46, 14, 6, 55, 1, 1, 77, 1, 1, 82, 0, 20, 92, 1, 2, 105, 31, 1, 122, 1, 1, 166, 2, 16, 168
Offset: 0

Views

Author

Gus Wiseman, Nov 01 2023

Keywords

Examples

			The a(n) partitions for n = 3, 9, 15, 21, 25, 27:
(3)  (9)      (15)         (21)             (25)         (27)
     (3,3,3)  (5,5,5)      (7,7,7)          (15,5,5)     (9,9,9)
              (9,3,3)      (9,9,3)          (5,5,5,5,5)  (15,9,3)
              (3,3,3,3,3)  (15,3,3)                      (21,3,3)
                           (9,3,3,3,3)                   (9,9,3,3,3)
                           (3,3,3,3,3,3,3)               (15,3,3,3,3)
                                                         (9,3,3,3,3,3,3)
                                                         (3,3,3,3,3,3,3,3,3)

Links

Crossrefs

Allowing even parts gives A018783, complement A000837.
For parts > 1 instead of gcd > 1 we have A087897.
For gcd = 1 instead of gcd > 1 we have A366843.
The strict case is A366750, with evens A303280.
The strict complement is A366844, with evens A078374.
A000041 counts integer partitions, strict A000009 (also into odd parts).
A000700 counts strict partitions into odd parts.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
A366842 counts partitions whose odd parts have a common divisor > 1.
Cf. A007359, A047967, A051424, A055922, A066208, A302698, A337485, A365067, A366845, A366848.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And@@OddQ/@#&&GCD@@#>1&]],{n,15}]
  • Python
    from math import gcd
from sympy.utilities.iterables import partitions
def A366852(n): return sum(1 for p in partitions(n) if all(d&1 for d in p) and gcd(*p)>1) # Chai Wah Wu, Nov 02 2023

Extensions

More terms from Chai Wah Wu, Nov 02 2023
a(0)=0 prepended by Alois P. Heinz, Jan 11 2024

A303365 Number of integer partitions of the n-th squarefree number using squarefree numbers.

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 28, 36, 60, 76, 96, 150, 228, 342, 416, 504, 877, 1484, 1759, 2079, 2885, 3387, 3968, 5413, 6304, 7328, 9852, 11395, 13159, 20082, 23056, 39532, 51385, 66488, 85660, 97078, 109907, 140465, 158573, 226918, 255268, 286920, 361606, 405470
Offset: 1

Views

Author

Gus Wiseman, Apr 22 2018

Keywords

Examples

			The a(5) = 9 partitions are (6), (51), (33), (321), (3111), (222), (2211), (21111), (111111).

Links

Crossrefs

Cf. A000051, A005117, A072774, A073576, A078374, A302491, A302796, A303283, A303364.

Programs

  • Mathematica
    nn=80;
sqf=Select[Range[nn],SquareFreeQ];
ser=Product[1/(1-x^sqf[[n]]),{n,Length[sqf]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,sqf}]

Formula

a(n) = A073576(A005117(n)).
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