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.

Showing 1-5 of 5 results.

A285798 Number of partitions of n into parts with an even number of distinct prime divisors.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 5, 5, 6, 7, 8, 8, 11, 11, 14, 16, 19, 19, 25, 26, 31, 34, 40, 41, 51, 53, 62, 68, 80, 85, 103, 107, 124, 135, 157, 166, 195, 205, 235, 256, 294, 311, 362, 383, 437, 472, 535, 568, 652, 695, 786, 847, 954, 1016, 1155, 1231, 1381, 1486, 1662, 1774, 1997, 2130, 2377, 2557, 2846
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 26 2017

Keywords

Examples

			a(10) = 3 because we have [10], [6, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A030231, A087153 (number of partitions into parts with an even number of divisors), A285799.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Product[1/(1 - Boole[EvenQ[PrimeNu[k]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - x^A030231(k)).

A286218 Number of partitions of n into parts with an odd number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 4, 4, 6, 7, 9, 11, 13, 16, 19, 23, 28, 33, 40, 46, 55, 65, 76, 89, 104, 121, 141, 163, 190, 219, 253, 290, 334, 383, 439, 502, 573, 653, 744, 845, 961, 1089, 1234, 1395, 1576, 1780, 2007, 2259, 2544, 2856, 3209, 3598, 4033, 4516, 5051, 5644, 6304, 7033, 7843
Offset: 0

Views

Author

Ilya Gutkovskiy, May 04 2017

Keywords

Examples

			a(8) = 4 because we have [8], [5, 3], [3, 3, 2] and [2, 2, 2, 2].

Links

Crossrefs

Cf. A001156, A026424, A285799, A286219.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      `if`(bigomega(d)::odd, d, 0), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, May 04 2017
  • Mathematica
    nmax = 60; CoefficientList[Series[Product[1/(1 - Boole[OddQ[PrimeOmega[k]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - x^A026424(k)).

A286220 Number of partitions of n into distinct parts with an odd number of distinct prime divisors.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 3, 2, 4, 3, 5, 5, 6, 7, 7, 10, 9, 12, 12, 15, 15, 18, 19, 22, 24, 26, 30, 32, 36, 40, 43, 49, 52, 58, 63, 69, 76, 81, 91, 96, 108, 114, 127, 135, 148, 159, 173, 186, 202, 217, 234, 253, 271, 293, 313, 339, 361, 390, 416, 449, 478, 514, 547, 588, 625, 671, 714, 763, 815, 867
Offset: 0

Views

Author

Ilya Gutkovskiy, May 04 2017

Keywords

Examples

			a(9) = 4 because we have [9], [7, 2], [5, 4] and [4, 3, 2].

Links

Crossrefs

Cf. A030230, A285799, A286221.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Product[1 + Boole[OddQ[PrimeNu[k]]] x^k, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A030230(k)).

A286224 Number of compositions (ordered partitions) of n into parts with an odd number of distinct prime divisors.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 4, 8, 11, 19, 28, 47, 72, 116, 182, 289, 460, 724, 1153, 1820, 2891, 4572, 7249, 11482, 18190, 28821, 45651, 72338, 114582, 181549, 287597, 455647, 721849, 1143590, 1811753, 2870247, 4547245, 7203933, 11412922, 18080907, 28644799, 45380602, 71894401, 113899027, 180444897, 285870668
Offset: 0

Views

Author

Ilya Gutkovskiy, May 04 2017

Keywords

Examples

			a(6) = 4 because we have [4, 2], [3, 3], [2, 4] and [2, 2, 2].

Links

Crossrefs

Cf. A030230, A285799, A286220, A286225.

Programs

  • Mathematica
    nmax = 45; CoefficientList[Series[1/(1 - Sum[Boole[OddQ[PrimeNu[k]]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=1} x^A030230(k)).

A327669 Sum of divisors of n that have an odd number of distinct prime factors.

Original entry on oeis.org

0, 2, 3, 6, 5, 5, 7, 14, 12, 7, 11, 9, 13, 9, 8, 30, 17, 14, 19, 11, 10, 13, 23, 17, 30, 15, 39, 13, 29, 40, 31, 62, 14, 19, 12, 18, 37, 21, 16, 19, 41, 54, 43, 17, 17, 25, 47, 33, 56, 32, 20, 19, 53, 41, 16, 21, 22, 31, 59, 104, 61, 33, 19, 126, 18, 82, 67, 23, 26, 84
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 21 2019

Keywords

Links

Crossrefs

Cf. A000203, A030230, A049060, A092248, A285799, A318677, A327670.

Programs

  • Maple
    with(numtheory):
a:= n-> add(`if`(nops(factorset(d))::odd, d, 0), d=divisors(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 21 2019
  • Mathematica
    a[n_] := DivisorSum[n, # &, OddQ[PrimeNu[#]] &]; Table[a[n], {n, 1, 70}]

Formula

G.f.: Sum_{k>=1} A030230(k) * x^A030230(k) / (1 - x^A030230(k)).
L.g.f.: log(B(x)) = Sum_{n>=1} a(n) * x^n / n, where B(x) = g.f. of A285799.
a(n) = Sum_{d|n} d * A092248(d).
a(n) = A000203(n) - A327670(n).
a(p) = p, where p is prime.
Showing 1-5 of 5 results.

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