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.

Previous Showing 51-57 of 57 results.

A329466 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} (1 + x^(k*j*(j + 1)/2))).

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 2, 2, 3, 3, 2, 4, 2, 3, 3, 4, 2, 5, 3, 5, 5, 4, 1, 6, 4, 3, 4, 7, 3, 7, 5, 7, 3, 5, 5, 8, 5, 6, 6, 8, 3, 10, 4, 7, 8, 7, 5, 10, 7, 10, 5, 10, 6, 9, 9, 13, 7, 8, 6, 14, 7, 10, 10, 14, 9, 12, 9, 12, 7, 17, 8, 14, 10, 14, 12, 17, 12, 12, 10, 20
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 13 2019

Keywords

Comments

Inverse Moebius transform of A024940.

Crossrefs

Cf. A024940, A047966, A047968, A329462, A329465.

Programs

  • Mathematica
    nmax = 80; CoefficientList[Series[Sum[-1 + Product[(1 + x^(k j (j + 1)/2)), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} A024940(k) * x^k / (1 - x^k).
a(n) = Sum_{d|n} A024940(d).

A302594 Numbers whose prime indices other than 1 are equal prime numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 16, 17, 18, 20, 22, 24, 25, 27, 31, 32, 34, 36, 40, 41, 44, 48, 50, 54, 59, 62, 64, 67, 68, 72, 80, 81, 82, 83, 88, 96, 100, 108, 109, 118, 121, 124, 125, 127, 128, 134, 136, 144, 157, 160, 162, 164, 166, 176, 179, 191, 192
Offset: 1

Views

Author

Gus Wiseman, Apr 10 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.

Examples

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
06: {{},{1}}
08: {{},{},{}}
09: {{1},{1}}
10: {{},{2}}
11: {{3}}
12: {{},{},{1}}
16: {{},{},{},{}}
17: {{4}}
18: {{},{1},{1}}
20: {{},{},{2}}
22: {{},{3}}
24: {{},{},{},{1}}

Crossrefs

Cf. A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A047968, A281113, A295924, A301760, A302242, A302243.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[400],MatchQ[Union[DeleteCases[primeMS[#],1]],{_?PrimeQ}|{}]&]

A320942 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} (1 + x^(k*j))/(1 - x^(k*j))).

Original entry on oeis.org

2, 6, 10, 20, 26, 54, 66, 120, 164, 262, 346, 572, 730, 1110, 1506, 2182, 2866, 4156, 5402, 7612, 9978, 13638, 17730, 24200, 31092, 41558, 53572, 70692, 90250, 118406, 150146, 194794, 246610, 316678, 398730, 509560, 637594, 808342, 1009186, 1270984, 1578530, 1978758, 2447066
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 24 2018

Keywords

Comments

Inverse Möbius transform of A015128.

Links

Crossrefs

Cf. A015128, A047966, A047968, A300274.

Programs

  • Maple
    a:=series(add(-1+mul((1+x^(k*j))/(1-x^(k*j)),j=1..100),k=1..100),x=0,44): seq(coeff(a,x,n),n=1..43); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 43; Rest[CoefficientList[Series[Sum[-1 + Product[(1 + x^(k j))/(1 - x^(k j)), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 43; Rest[CoefficientList[Series[Sum[1/EllipticTheta[4, 0, x^k] - 1, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[Sum[PartitionsP[d - k] PartitionsQ[k], {k, 0, d}], {d, Divisors[n]}], {n, 43}]

Formula

G.f.: Sum_{k>=1} A015128(k)*x^k/(1 - x^k).
G.f.: Sum_{k>=1} (1/theta_4(x^k) - 1), where theta_4() is the Jacobi theta function.
a(n) = Sum_{d|n} A015128(d).

A329434 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} (1 + x^(k*(2*j - 1)))).

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 4, 4, 4, 3, 7, 4, 5, 7, 9, 6, 10, 7, 12, 11, 11, 10, 20, 14, 16, 18, 22, 18, 28, 21, 32, 29, 32, 32, 47, 36, 44, 46, 60, 50, 67, 58, 75, 77, 82, 79, 112, 95, 114, 114, 134, 126, 157, 148, 181, 176, 196, 193, 248, 224, 257, 268, 308, 299
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 13 2019

Keywords

Comments

Inverse Moebius transform of A000700.

Crossrefs

Cf. A000700, A047966, A047968.

Programs

  • Mathematica
    nmax = 65; CoefficientList[Series[Sum[-1 + Product[(1 + x^(k (2 j - 1))), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} (-1 + Product_{j>=1} 1 / (1 + (-1)^j * x^(k*j))).
G.f.: Sum_{k>=1} A000700(k) * x^k / (1 - x^k).
a(n) = Sum_{d|n} A000700(d).

A333697 a(n) = Sum_{d|n} (-1)^omega(n/d) * phi(rad(n/d)) * p(d), where p = A000041 (partition numbers).

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 9, 14, 22, 31, 46, 59, 89, 114, 158, 201, 281, 337, 472, 570, 756, 936, 1233, 1456, 1926, 2323, 2942, 3556, 4537, 5334, 6812, 8088, 10021, 11997, 14805, 17432, 21601, 25507, 30971, 36606, 44543, 52106, 63219, 74097, 88680, 104281, 124708, 145205, 173429, 202124
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 02 2020

Keywords

Crossrefs

Cf. A000010, A000041, A001221, A007947, A008683, A023900, A047968, A078392.

Programs

  • Mathematica
    Table[Sum[(-1)^PrimeNu[n/d] EulerPhi[Last[Select[Divisors[n/d], SquareFreeQ]]] PartitionsP[d], {d, Divisors[n]}], {n, 50}]
  • PARI
    rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = sumdiv(n, d, (-1)^omega(n/d) * eulerphi(rad(n/d)) * numbpart(d)); \\ Michel Marcus, Apr 03 2020

Formula

a(n) = Sum_{d|n} A023900(n/d) * A000041(d).
a(n) = Sum_{d|n} A047968(n/d) * mu(d) * d.
Sum_{k=1..n} a(gcd(n,k)) = A000041(n).

A336129 Number of strict compositions of divisors of n.

Original entry on oeis.org

1, 2, 4, 5, 6, 16, 14, 24, 31, 64, 66, 120, 134, 208, 360, 459, 618, 894, 1178, 1622, 2768, 3364, 4758, 6432, 8767, 11440, 15634, 24526, 30462, 42296, 55742, 75334, 98112, 131428, 168444, 258403, 315974, 432244, 558464, 753132, 958266, 1280840, 1621274
Offset: 1

Views

Author

Gus Wiseman, Jul 11 2020

Keywords

Comments

A strict composition of k is a finite sequence of distinct positive integers summing to k.

Examples

			The a(1) = 1 through a(7) = 14 compositions:
  (1)  (1)  (1)    (1)    (1)    (1)      (1)
       (2)  (3)    (2)    (5)    (2)      (7)
            (1,2)  (4)    (1,4)  (3)      (1,6)
            (2,1)  (1,3)  (2,3)  (6)      (2,5)
                   (3,1)  (3,2)  (1,2)    (3,4)
                          (4,1)  (1,5)    (4,3)
                                 (2,1)    (5,2)
                                 (2,4)    (6,1)
                                 (4,2)    (1,2,4)
                                 (5,1)    (1,4,2)
                                 (1,2,3)  (2,1,4)
                                 (1,3,2)  (2,4,1)
                                 (2,1,3)  (4,1,2)
                                 (2,3,1)  (4,2,1)
                                 (3,1,2)
                                 (3,2,1)

Crossrefs

Compositions of divisors are A034729.
Strict partitions of divisors are A047966.
Partitions of divisors are A047968.
Cf. A000005, A001970, A006951, A133494, A318683, A336128, A336130, A336132.

Programs

  • Mathematica
    Table[Sum[Length[Join@@Permutations/@Select[IntegerPartitions[d],UnsameQ@@#&]],{d,Divisors[n]}],{n,12}]

Formula

Moebius transform is A032020 (strict compositions).

A387116 Number of ways to choose a constant sequence of integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 0, 5, 1, 2, 0, 7, 0, 11, 0, 0, 1, 15, 0, 22, 0, 0, 0, 30, 0, 3, 0, 2, 0, 42, 0, 56, 1, 0, 0, 0, 0, 77, 0, 0, 0, 101, 0, 135, 0, 0, 0, 176, 0, 5, 0, 0, 0, 231, 0, 0, 0, 0, 0, 297, 0, 385, 0, 0, 1, 0, 0, 490, 0, 0, 0, 627, 0, 792, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Aug 21 2025

Keywords

Comments

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.
If n is a prime power prime(x)^y, then a(n) is the number of integer partitions of x; otherwise, a(n) = 0.

Examples

			The a(49) = 5 choices:
  ((4),(4))
  ((3,1),(3,1))
  ((2,2),(2,2))
  ((2,1,1),(2,1,1))
  ((1,1,1,1),(1,1,1,1))

Crossrefs

Positions of zeros are A024619, complement A000961.
Twice-partitions of this type are counted by A047968, see also A296122.
For initial intervals instead of partitions we have A055396, see also A387111.
This is the constant case of A299200, see also A357977, A357982.
For disjoint instead of constant we have A383706.
For distinct instead of constant we have A387110.
For divisors instead of partitions we have A387114, see also A355731, A355739.
For strict partitions instead of partitions we have A387117.
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. A008284, A063834, A276078, A335433, A335448, A355529, A355741, A387115, A387120, A387133, A387136.

Programs

Formula

a(n) = A000041(A297109(n)).
Previous Showing 51-57 of 57 results.

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