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 11-17 of 17 results.

A347777 Number of compositions (ordered partitions) of n into at most 2 squarefree parts.

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 4, 5, 6, 4, 4, 5, 7, 7, 6, 7, 10, 9, 8, 7, 11, 9, 10, 9, 14, 10, 10, 10, 13, 11, 10, 11, 16, 13, 14, 13, 22, 15, 14, 15, 22, 17, 16, 19, 25, 20, 16, 17, 26, 20, 16, 15, 27, 21, 20, 15, 26, 21, 22, 19, 29, 23, 22, 22, 30, 23, 22, 23, 35, 25, 26
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 13 2021

Keywords

Crossrefs

Cf. A005117, A098235, A280194, A347778, A347779, A347780.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,2,Select[Range@n,SquareFreeQ]],1],{n,0,100}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

A307815 Number of partitions of n into 3 squarefree parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 3, 5, 4, 5, 5, 7, 7, 9, 8, 11, 11, 13, 11, 15, 14, 18, 15, 20, 19, 23, 20, 24, 24, 27, 24, 30, 29, 34, 30, 37, 36, 42, 36, 45, 44, 50, 44, 54, 54, 59, 52, 62, 63, 68, 57, 69, 70, 78, 65, 78, 78, 88, 74, 86, 87, 98, 84, 98, 98, 107, 93, 109, 108, 120, 102, 124, 123
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 30 2019

Keywords

Examples

			a(10) = 4 because we have [7, 2, 1], [6, 3, 1], [6, 2, 2] and [5, 3, 2].

Links

Crossrefs

Cf. A005117, A008683, A068307, A071068, A073576, A098235, A280210, A307727.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1, 0$3], `if`(i<1, 0, b(n, i-1)+
      `if`(numtheory[issqrfree](i), [0, b(n-i, min(i, n-i))[1..3][]], 0)))
    end:
a:= n-> b(n$2)[4]:
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 30 2019
  • Mathematica
    Array[Count[IntegerPartitions[#, {3}], _?(AllTrue[#, SquareFreeQ] &)] &, 75, 0]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, {1, 0, 0, 0}, If[i < 1, {0, 0, 0, 0}, b[n, i - 1] + If[SquareFreeQ[i], {0, Sequence @@ b[n - i, Min[i, n - i]][[1 ;; 3]]}, {0, 0, 0, 0}]]];
a[n_] := b[n, n][[4]];
a /@ Range[0, 80] (* Jean-François Alcover, Jun 06 2021, after Alois P. Heinz *)

Formula

a(n) = [x^n y^3] Product_{k>=1} 1/(1 - mu(k)^2*y*x^k).
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} mu(i)^2 * mu(k)^2 * mu(n-i-k)^2, where mu is the Mobius function. - Wesley Ivan Hurt, May 09 2019

A280683 Number of ways to write n as an ordered sum of two positive squarefree semiprimes (A006881).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 2, 0, 2, 3, 2, 1, 2, 4, 0, 0, 2, 6, 2, 0, 2, 4, 4, 1, 4, 5, 4, 0, 4, 8, 6, 2, 0, 5, 4, 4, 4, 6, 4, 0, 4, 8, 10, 0, 2, 4, 6, 3, 6, 9, 4, 3, 6, 14, 8, 2, 4, 5, 8, 3, 10, 8, 4, 0, 8, 12, 4, 4, 4, 8, 6, 8, 12, 11, 6, 2, 10, 12, 12, 4, 8, 12, 12, 5, 12, 10, 4, 6
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 07 2017

Keywords

Comments

Conjecture: a(n) > 0 for n > 82 (see comment in A006881 from Richard R. Forberg).

Examples

			a(20) = 3 because we have [14, 6], [10, 10] and [6, 14].

Links

Crossrefs

Cf. A001222, A001358, A005117, A006881, A008683, A073610, A098235, A199331, A280710.

Programs

  • Mathematica
    nmax = 106; Rest[CoefficientList[Series[(Sum[MoebiusMu[k]^2 Floor[PrimeOmega[k]/2] Floor[2/PrimeOmega[k]] x^k, {k, 2, nmax}])^2, {x, 0, nmax}], x]]

Formula

G.f.: (Sum_{k>=2} mu(k)^2*floor(bigomega(k)/2)*floor(2/bigomega(k))*x^k)^2, where mu(k) is the Moebius function (A008683) and bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).
a(n) = Sum_{k=1..n-1} c(k) * c(n-k), where c = A280710. - Wesley Ivan Hurt, Jan 07 2024

A285796 Number of ways to write n as an ordered sum of two numbers that are the product of an even number of distinct primes (including 1).

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 1, 0, 0, 2, 4, 0, 0, 0, 3, 2, 2, 2, 2, 2, 0, 4, 3, 2, 1, 2, 4, 0, 2, 4, 8, 2, 0, 4, 6, 4, 1, 4, 5, 4, 0, 6, 8, 6, 2, 0, 7, 4, 4, 4, 8, 4, 2, 6, 8, 10, 0, 4, 4, 6, 5, 6, 9, 4, 5, 6, 14, 8, 2, 6, 5, 8, 5, 10, 8, 4, 0, 10, 12, 4, 6, 6, 10, 6, 8, 12, 13, 6, 4, 12, 14, 12, 4, 8, 12
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 26 2017

Keywords

Comments

Conjecture: a(n) > 0 for all n > 82.

Examples

			a(16) = 4 because we have [15, 1], [10, 6], [6, 10] and [1, 15].

Crossrefs

Cf. A005117, A030229, A098235, A285797.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[(Sum[Boole[MoebiusMu[k] == 1] x^k, {k, 1, nmax}])^2, {x, 0, nmax}], x]

Formula

G.f.: (Sum_{k>=1} x^A030229(k))^2.

A285797 Number of ways to write n as an ordered sum of two numbers that are the product of an odd number of distinct primes.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 2, 2, 2, 3, 0, 2, 2, 3, 2, 4, 0, 4, 2, 4, 2, 5, 0, 6, 2, 5, 0, 4, 0, 6, 2, 6, 4, 7, 2, 8, 2, 3, 2, 6, 2, 8, 4, 8, 4, 7, 4, 10, 6, 8, 0, 6, 4, 10, 4, 6, 0, 7, 4, 13, 6, 5, 2, 10, 2, 12, 2, 6, 4, 10, 6, 16, 10, 9, 4, 10, 6, 14, 4, 10, 6, 9, 10, 17, 8, 9, 2, 8, 10, 18, 6, 8, 2, 9, 6, 16, 6, 6, 4, 14
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 26 2017

Keywords

Comments

Conjecture: a(n) > 0 for all n > 57.

Examples

			a(10) = 3 because we have [7, 3], [5, 5] and [3, 7].

Crossrefs

Cf. A005117, A030059, A098235, A285796.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[(Sum[Boole[MoebiusMu[k] == -1] x^k, {k, 1, nmax}])^2, {x, 0, nmax}], x]

Formula

G.f.: (Sum_{k>=1} x^A030059(k))^2.

A290136 Positive numbers that are not the sum of two nonprime squarefree numbers (A000469).

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 9, 10, 13, 14, 17, 18, 19, 26, 33, 38, 46, 62, 82
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 20 2017

Keywords

Comments

The sequence is conjectured to be complete.

Crossrefs

Cf. A000469, A014092, A071068, A071331, A098235, A239508, A239509.

Programs

  • Mathematica
    nmax = 82; f[x_] := Sum[Boole[SquareFreeQ[k] && PrimeNu[k] != 1] x^k, {k, 1, nmax}]^2; b = Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]; c = Complement[Range[nmax], b][[1 ;; 19]]

A286971 Number of ways to write n as a sum of two numbers, one of which is the product of an even number of distinct primes (including 1) (A030229) and another is the product of an odd number of distinct primes (A030059).

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 1, 2, 2, 1, 1, 1, 4, 2, 2, 2, 2, 1, 3, 3, 3, 2, 3, 3, 4, 1, 3, 3, 4, 2, 3, 3, 5, 5, 4, 5, 5, 3, 5, 6, 6, 4, 3, 4, 4, 3, 7, 7, 6, 3, 3, 6, 8, 6, 4, 4, 3, 8, 8, 8, 7, 2, 7, 10, 8, 5, 5, 6, 4, 8, 8, 12, 7, 3, 7, 11, 11, 8, 3, 7, 9, 6, 10, 14, 8, 4, 5, 12, 13, 10, 7, 9, 8, 12, 13, 12
Offset: 0

Views

Author

Ilya Gutkovskiy, May 17 2017

Keywords

Comments

Conjecture: a(n) > 0 for all n > 10.

Examples

			a(17) = 4 because we have [15, 2], [14, 3], [11, 6] and [10, 7].

Crossrefs

Cf. A005117, A030059, A030229, A098235, A098236, A285796, A285797.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[(Sum[Boole[MoebiusMu[k] == 1] x^k, {k, 1, nmax}]) (Sum[Boole[MoebiusMu[k] == -1] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: (Sum_{i>=1} x^A030229(i))*(Sum_{j>=1} x^A030059(j)).
Previous Showing 11-17 of 17 results.

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