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.

A339381 Number of partitions of n into an odd number of primes (counting 1 as a prime).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 3, 7, 5, 11, 9, 18, 14, 27, 22, 40, 33, 58, 48, 82, 69, 114, 97, 157, 134, 212, 183, 284, 246, 376, 327, 493, 431, 640, 562, 825, 728, 1056, 934, 1341, 1191, 1694, 1508, 2126, 1899, 2654, 2377, 3297, 2960, 4075, 3668, 5015, 4523, 6145, 5554, 7499
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

Examples

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

Links

Crossrefs

Cf. A000607, A008578, A027193, A034891, A184171, A184172, A184198, A184199, A338826, A339380, A339382, A339383.

Programs

  • Maple
    b:= proc(n, i, t) option remember; (p->
      `if`(n=0, t, `if`(i<0, 0, b(n, i-1, t)+
      `if`(p>n, 0, b(n-p, i, 1-t)))))(`if`(i<1, 1, ithprime(i)))
    end:
a:= n-> b(n, numtheory[pi](n), 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 02 2020
  • Mathematica
    nmax = 55; CoefficientList[Series[(1/2) ((1/(1 - x)) Product[1/(1 - x^Prime[k]), {k, 1, nmax}] - (1/(1 + x)) Product[1/(1 + x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * ((1/(1 - x)) * Product_{k>=1} 1 / (1 - x^prime(k)) - (1/(1 + x)) * Product_{k>=1} 1 / (1 + x^prime(k))).
a(n) = (A034891(n) - A338826(n)) / 2.

A339382 Number of partitions of n into an even number of distinct primes (counting 1 as a prime).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 4, 4, 4, 4, 6, 5, 5, 5, 6, 6, 7, 7, 9, 8, 9, 8, 11, 10, 11, 12, 14, 12, 15, 14, 17, 16, 17, 17, 22, 20, 22, 21, 25, 24, 28, 27, 31, 30, 33, 31, 39, 36, 40, 40, 46, 42, 49, 47, 54, 53, 58, 55, 67, 63, 70, 68
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

Examples

			a(16) = 3 because we have [13, 3], [11, 5] and [7, 5, 3, 1].

Links

Crossrefs

Cf. A000586, A008578, A036497, A067661, A184171, A184172, A184198, A184199, A298602, A339380, A339381, A339383.

Programs

  • Maple
    s:= proc(n) option remember;
      `if`(n<1, n+1, ithprime(n)+s(n-1))
    end:
b:= proc(n, i, t) option remember; (p-> `if`(n=0, t,
      `if`(n>s(i), 0, b(n, i-1, t)+ `if`(p>n, 0,
       b(n-p, i-1, 1-t)))))(`if`(i<1, 1, ithprime(i)))
    end:
a:= n-> b(n, numtheory[pi](n), 1):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 02 2020
  • Mathematica
    nmax = 75; CoefficientList[Series[(1/2) ((1 + x) Product[(1 + x^Prime[k]), {k, 1, nmax}] + (1 - x) Product[(1 - x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * ((1 + x) * Product_{k>=1} (1 + x^prime(k)) + (1 - x) * Product_{k>=1} (1 - x^prime(k))).
a(n) = (A036497(n) + A298602(n)) / 2.

A339383 Number of partitions of n into an odd number of distinct primes (counting 1 as a prime).

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 6, 6, 5, 7, 6, 8, 7, 8, 9, 10, 9, 12, 11, 12, 11, 14, 14, 16, 15, 17, 17, 20, 17, 21, 22, 24, 22, 27, 25, 30, 28, 31, 31, 36, 33, 40, 39, 42, 40, 47, 46, 53, 49, 55, 54, 63, 58, 68, 67, 73
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

Examples

			a(21) = 4 because we have [17, 3, 1], [13, 7, 1], [13, 5, 3] and [11, 7, 3].

Links

Crossrefs

Cf. A000586, A008578, A036497, A067659, A184171, A184172, A184198, A184199, A298602, A339380, A339381, A339382.

Programs

  • Maple
    s:= proc(n) option remember;
      `if`(n<1, n+1, ithprime(n)+s(n-1))
    end:
b:= proc(n, i, t) option remember; (p-> `if`(n=0, t,
      `if`(n>s(i), 0, b(n, i-1, t)+ `if`(p>n, 0,
       b(n-p, i-1, 1-t)))))(`if`(i<1, 1, ithprime(i)))
    end:
a:= n-> b(n, numtheory[pi](n), 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 02 2020
  • Mathematica
    nmax = 75; CoefficientList[Series[(1/2) ((1 + x) Product[(1 + x^Prime[k]), {k, 1, nmax}] - (1 - x) Product[(1 - x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * ((1 + x) * Product_{k>=1} (1 + x^prime(k)) - (1 - x) * Product_{k>=1} (1 - x^prime(k))).
a(n) = (A036497(n) - A298602(n)) / 2.

A339408 Number of compositions (ordered partitions) of n into an even number of primes.

Original entry on oeis.org

1, 0, 0, 0, 1, 2, 1, 2, 3, 6, 9, 8, 16, 24, 40, 52, 72, 112, 172, 256, 364, 528, 804, 1188, 1757, 2548, 3782, 5614, 8308, 12214, 17979, 26586, 39352, 58044, 85608, 126248, 186630, 275556, 406737, 600066, 885952, 1308250, 1931473, 2850692, 4207952, 6212110, 9171800, 13538980
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2020

Keywords

Examples

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

Links

Crossrefs

Cf. A000040, A023360, A034008, A184198, A339380, A339409.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, t, add(
      b(n-ithprime(j), 1-t), j=1..numtheory[pi](n)))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..55);  # Alois P. Heinz, Dec 03 2020
  • Mathematica
    nmax = 47; CoefficientList[Series[(1/2) (1/(1 - Sum[x^Prime[k], {k, 1, nmax}]) + 1/(1 + Sum[x^Prime[k], {k, 1, nmax}])), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^prime(k)) + 1 / (1 + Sum_{k>=1} x^prime(k))).

A339395 Number of partitions of n into an even number of nonprime parts.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 4, 6, 7, 8, 10, 13, 14, 19, 20, 26, 29, 36, 40, 51, 56, 70, 76, 96, 105, 129, 143, 172, 192, 231, 254, 308, 339, 402, 447, 529, 586, 691, 764, 896, 993, 1159, 1281, 1493, 1652, 1912, 2114, 2445, 2699, 3110, 3436, 3939, 4356, 4982, 5497, 6280
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

Examples

			a(9) = 3 because we have [8, 1], [6, 1, 1, 1] and [4, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A002095, A018252, A027187, A184198, A302236, A339380, A339396.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[(1/2) (Product[(1 - x^Prime[k])/(1 - x^k), {k, 1, nmax}] + Product[(1 + x^Prime[k])/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=1} (1 - x^prime(k)) / (1 - x^k) + Product_{k>=1} (1 + x^prime(k)) / (1 + x^k)).
a(n) = (A002095(n) + A302236(n)) / 2.
Showing 1-5 of 5 results.

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