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.

A279615 Partial sums of A279315.

Original entry on oeis.org

0, 0, 1, 3, 7, 9, 10, 10, 16, 16, 17, 29, 30, 30, 42, 42, 43, 49, 50, 50, 52, 52, 53, 53, 53, 55, 55, 55, 56, 86, 87, 87, 87, 89, 89, 89, 90, 90, 92, 92, 93, 105, 106, 106, 108, 108, 109, 109, 109, 111, 111, 111, 115, 115, 115, 117, 117, 117, 118, 124, 125, 125, 125, 127, 127, 127, 128, 128, 130, 130, 131, 133, 134
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 15 2016

Keywords

Links

Crossrefs

Cf. A010051, A279315.

Programs

  • Maple
    with(numtheory): a:=n->add(add( (pi(i)-pi(i-1)) * (pi(2*h-i)-pi(2*h-i-1)) * (pi(2*h-i)-pi(i-1)) * (product(1-abs((pi(k)-pi(k-1))-(pi(2*h-k)-pi(2*h-k-1))), k=i..h)), i=3..h), h=1..n): seq(a(n), n=1..80);

Formula

a(n) = Sum_{h=1..n} (Sum_{i=3..h} A010051(i) * A010051(2h-i) * (pi(2h-i)-pi(i-1)) * (Product_{k=i..h} 1-abs(A010051(k)-A010051(2h-k)))).

A279727 Sum of the smaller parts of the Goldbach partitions (p,q) of 2n such that all primes from p to q (inclusive) appear as a part in some Goldbach partition of p+q = 2n.

Original entry on oeis.org

0, 0, 3, 3, 8, 5, 7, 0, 12, 0, 11, 23, 13, 0, 31, 0, 17, 30, 19, 0, 19, 0, 23, 0, 0, 23, 0, 0, 29, 101, 31, 0, 0, 31, 0, 0, 37, 0, 37, 0, 41, 109, 43, 0, 43, 0, 47, 0, 0, 47, 0, 0, 100, 0, 0, 53, 0, 0, 59, 112, 61, 0, 0, 61, 0, 0, 67, 0, 67, 0, 71, 71, 73, 0, 0, 73, 0, 0
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 17 2016

Keywords

Links

Crossrefs

Cf. A010051, A279315, A279728, A279729.

Programs

  • Maple
    with(numtheory): A279727:=n->add( i * (pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)) * (product(1-abs((pi(k)-pi(k-1))-(pi(2*n-k)-pi(2*n-k-1))), k=i..n)), i=3..n): seq(A279727(n), n=1..100);
  • Mathematica
    Table[Sum[(i Boole[PrimeQ@ i] Boole[PrimeQ[2 n - i]]) Product[1 - Abs[Boole[PrimeQ@ k] - Boole[PrimeQ[2 n - k]]], {k, i, n}], {i, 3, n}], {n, 100}] (* Michael De Vlieger, Dec 18 2016 *)

Formula

a(n) = Sum_{i=3..n} (i * c(i) * c(2n-i) * (Product_{k=i..n} (1-abs(c(k)-c(2n-k))))), where c = A010051.

A279728 Sum of the larger parts of the Goldbach partitions (p,q) of 2n such that all primes from p to q (inclusive) appear as a part in some Goldbach partition of p+q = 2n.

Original entry on oeis.org

0, 0, 3, 5, 12, 7, 7, 0, 24, 0, 11, 49, 13, 0, 59, 0, 17, 42, 19, 0, 23, 0, 23, 0, 0, 29, 0, 0, 29, 199, 31, 0, 0, 37, 0, 0, 37, 0, 41, 0, 41, 143, 43, 0, 47, 0, 47, 0, 0, 53, 0, 0, 112, 0, 0, 59, 0, 0, 59, 128, 61, 0, 0, 67, 0, 0, 67, 0, 71, 0, 71, 73, 73, 0, 0, 79, 0, 0
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 17 2016

Keywords

Links

Crossrefs

Cf. A010051, A279315, A279727, A279729.

Programs

  • Maple
    with(numtheory): A279728:=n->add( (2*n-i) * (pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)) * (product(1-abs((pi(k)-pi(k-1))-(pi(2*n-k)-pi(2*n-k-1))), k=i..n)), i=3..n): seq(A279728(n), n=1..100);
  • Mathematica
    Table[Sum[((2 n - i) Boole[PrimeQ@ i] Boole[PrimeQ[2 n - i]]) Product[1 - Abs[Boole[PrimeQ@ k] - Boole[PrimeQ[2 n - k]]], {k, i, n}], {i, 3, n}], {n, 100}] (* Michael De Vlieger, Dec 18 2016 *)

Formula

a(n) = Sum_{i=3..n} ((2n-i) * c(i) * c(2n-i) * (Product_{k=i..n} (1-abs(c(k) - c(2n-k))))), where c = A010051.

A279536 Count the squarefree numbers appearing in each interval [p,q] where (p,q) is a Goldbach partition of 2n such that all primes from p to q (inclusive) appear as a part in some Goldbach partition of p+q = 2n, and then add the results.

Original entry on oeis.org

0, 0, 1, 2, 5, 3, 1, 0, 9, 0, 1, 19, 1, 0, 21, 0, 1, 10, 1, 0, 4, 0, 1, 0, 0, 3, 0, 0, 1, 68, 1, 0, 0, 5, 0, 0, 1, 0, 4, 0, 1, 25, 1, 0, 3, 0, 1, 0, 0, 3, 0, 0, 8, 0, 0, 5, 0, 0, 1, 12, 1, 0, 0, 5, 0, 0, 1, 0, 4, 0, 1, 2, 1, 0, 0, 5, 0, 0, 1, 0, 14, 0, 1, 0, 0, 5, 0, 0, 1, 0
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 14 2016

Keywords

Comments

a(n) >= A279315(n). - Wesley Ivan Hurt, Dec 17 2016

Links

Crossrefs

Cf. A008683, A010051, A279315.

Programs

  • Maple
    with(numtheory): A279536:=n->add( (pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)) * add(mobius(j)^2, j=i..2*n-i) * (product(1-abs((pi(k)-pi(k-1))-(pi(2*n-k)-pi(2*n-k-1))), k=i..n)), i=3..n): seq(A279536(n), n=1..100);

Formula

a(n) = Sum_{i=3..n} (A010051(i) * A010051(2n-i) * (Sum_{j=i..2n-i} mu(j)^2) * (Product_{k=i..n} (1-abs(A010051(k)-A010051(2n-k))))), where mu is the Möbius function (A008683).

A293909 Number of Goldbach partitions (p,q) of 2n, p <= q, such that both 2n-2 and 2n+2 have a Goldbach partition with a greater difference between its prime parts than q-p.

Original entry on oeis.org

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

Views

Author

Wesley Ivan Hurt, Oct 19 2017

Keywords

Examples

			a(9) = 2; Both 2(9)-2 = 16 and 2(9)+2 = 20 have two Goldbach partitions: 16 = 13+3 = 11+5 and 20 = 17+3 = 13+7. Note that 13-3 = 10 and 17-3 = 14 are the largest differences of the primes among the Goldbach partitions of 2n-2 and 2n+2. The Goldbach partitions of 2(9) = 18 are 13+5 = 11+7. Since 13-5 = 8 and 11-7 = 4 are both less than min(10,14) = 10, a(9) = 2.

Links

Crossrefs

Cf. A002375, A226237, A278700, A279103, A279315, A279481, A279727, A279728, A279729, A279792.

Extensions

More terms from Bert Dobbelaere, Sep 15 2019
Showing 1-5 of 5 results.

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