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 1-3 of 3 results.

A281274 Expansion of Product_{j>=1} (1 + x^(Sum_{i=1..j} prime(i))).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 2, 0, 2, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 3, 0, 3, 0, 1, 2, 0, 2, 0, 0, 2, 1, 2, 1, 0, 2, 1, 3, 1, 2, 0, 2, 1, 1, 2, 0, 2, 1, 3, 2, 2, 1, 1, 2, 2, 2, 2, 0, 3, 0, 2, 2, 1, 4, 1, 3, 2, 3, 2, 2, 1, 2, 3
Offset: 0

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Author

Ilya Gutkovskiy, Jan 18 2017

Keywords

Comments

Number of partitions of n into distinct nonzero partial sums of primes (A007504).

Examples

			a(17) = 2 because we have [17] and [10, 5, 2], where 2 = prime(1), 5 = prime(1) + prime(2), 10 = prime(1) + prime(2) + prime(3), 17 = prime(1) + prime(2) + prime(3) + prime(4).

Links

Crossrefs

Cf. A000586, A000607, A007504, A281273.

Programs

  • Mathematica
    nmax = 110; CoefficientList[Series[Product[1 + x^Sum[Prime[i], {i, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{j>=1} (1 + x^(Sum_{i=1..j} prime(i))).

A282970 Number of partitions of n into primes of form x^2 + y^2 (A002313).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 8, 9, 9, 10, 10, 12, 12, 13, 14, 14, 17, 16, 19, 19, 21, 22, 23, 25, 27, 27, 30, 30, 34, 35, 37, 40, 41, 45, 46, 50, 52, 55, 58, 60, 65, 67, 71, 75, 78, 84, 86, 92, 97, 100, 108, 110, 118, 123, 127, 137, 139, 150, 154, 162
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 25 2017

Keywords

Comments

Number of partitions of n into primes congruent to 1 or 2 mod 4.

Examples

			a(10) = 2 because we have [5, 5] and [2, 2, 2, 2, 2].

Links

Crossrefs

Cf. A000607, A002313, A024941, A281273.

Programs

  • Mathematica
    nmax = 82; CoefficientList[Series[Product[1/(1 - Boole[SquaresR[2, k] != 0 && PrimeQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
  • PARI
    Vec(prod(k=1, 82, (1/(1 - (isprime(k) && k%4<3)*x^k))) + O(x^83)) \\ Indranil Ghosh, Mar 15 2017

Formula

G.f.: Product_{k>=1} 1/(1 - x^A002313(k)).

A282906 Expansion of 1/(1 - Sum_{j>=1} x^(Sum_{i=1..j} prime(i))).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 3, 4, 6, 5, 10, 9, 15, 18, 21, 32, 33, 52, 58, 79, 102, 122, 172, 201, 277, 341, 438, 575, 707, 947, 1169, 1530, 1949, 2474, 3228, 4046, 5281, 6678, 8594, 11035, 14025, 18142, 23015, 29681, 37888, 48512, 62319, 79456, 102230, 130456, 167418, 214356, 274221, 351904, 449700, 577024, 738150
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 24 2017

Keywords

Comments

Number of compositions (ordered partitions) of n into partial sums of primes (A007504).
Conjecture: every number > 3 is the sum of at most 5 partial sums of primes.

Examples

			a(11) = 4 because we have [5, 2, 2, 2], [2, 5, 2, 2], [2, 2, 5, 2] and [2, 2, 2, 5].

Links

Crossrefs

Cf. A007504, A023360, A084143, A281273, A281274.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[1/(1 - Sum[x^Sum[Prime[i], {i, 1, j}], {j, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{j>=1} x^(Sum_{i=1..j} prime(i))).
Showing 1-3 of 3 results.

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