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 1-10 of 14 results. Next

A352240 Even numbers with at least one pair of Goldbach partitions, (p,q) and (r,s) with p,q,r,s prime and p < r <= s < q, such that all integers in the open intervals (p,r) and (s,q) are composite.

Original entry on oeis.org

10, 16, 18, 22, 24, 30, 34, 36, 42, 46, 48, 54, 60, 64, 66, 72, 76, 78, 82, 84, 90, 96, 98, 102, 106, 108, 110, 112, 114, 120, 126, 132, 136, 138, 140, 142, 144, 150, 154, 156, 160, 162, 168, 174, 180, 184, 186, 188, 190, 192, 194, 196, 198, 202, 204, 210, 216, 218, 220, 222
Offset: 1

Views

Author

Wesley Ivan Hurt, Mar 08 2022

Keywords

Comments

Similar to A187797 but also contains the numbers 82, 96, 98, 110, 136, ...

Examples

			82 is in the sequence since it has at least one pair of Goldbach partitions, namely (23,59) and (29,53), such that all integers in the open intervals (23,29) and (53,59) are composite.

Links

Crossrefs

Cf. A187797, A278700, A352248, A352283.
Cf. A352442, A352443, A352444, A352445.

Programs

  • Mathematica
    Table[If[Sum[Sum[KroneckerDelta[NextPrime[k], i]*KroneckerDelta[NextPrime[2 n - i], 2 n - k]*(PrimePi[k] - PrimePi[k - 1]) (PrimePi[2 n - k] - PrimePi[2 n - k - 1]) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {k, i}], {i, n}] > 0, 2 n, {}], {n, 150}] // Flatten

Formula

a(n) = A352442(n) + A352443(n).
a(n) = A352444(n) + A352445(n).

A352248 Number of pairs of Goldbach partitions of A352240(n), (p,q) and (r,s) with p,q,r,s prime and p < r <= s < q, such that all integers in the open intervals (p,r) and (s,q) are composite.

Original entry on oeis.org

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

Views

Author

Wesley Ivan Hurt, Mar 09 2022

Keywords

Examples

			a(13) = 4; The Goldbach partitions of A352240(13) = 60 are: 7+53 = 13+47 = 17+43 = 19+41 = 23+37 = 29+31. The 4 pairs of Goldbach partitions of 60 that are being counted are: (13,47),(17,43); (17,43),(19,41); (19,41),(23,37); and (23,37),(29,31). Note that the pair (7,53),(13,47) is not counted since there is a prime in the interval (7,13), namely 11.

Links

Crossrefs

Cf. A187797, A278700, A352240, A352283.

Programs

  • Mathematica
    a[n_] := Sum[Sum[KroneckerDelta[NextPrime[k], i]*KroneckerDelta[NextPrime[2 n - i], 2 n - k]*(PrimePi[k] - PrimePi[k - 1]) (PrimePi[2 n - k] - PrimePi[2 n - k - 1]) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {k, i}], {i, n}];
Table[If[a[n] > 0, a[n], {}], {n, 100}] // Flatten

A352283 Smallest nonnegative even integer with exactly n pairs of Goldbach partitions, (p,q) and (r,s) with p,q,r,s prime and p < r <= s < q, such that all integers in the open intervals (p,r) and (s,q) are composite.

Original entry on oeis.org

0, 10, 24, 48, 60, 126, 90, 114, 120, 594, 240, 462, 300, 390, 210, 330, 510
Offset: 0

Views

Author

Wesley Ivan Hurt, Mar 10 2022

Keywords

Examples

			a(4) = 60 is the smallest nonnegative even integer with exactly 4 pairs of Goldbach partitions (13,47),(17,43); (17,43),(19,41); (19,41),(23,37); and (23,37),(29,31) with all integers composite in the open intervals: (13,17) and (43,47), (17,19) and (41,43), (19,23) and (37,41), (23,29) and (31,37) respectively.

Links

Crossrefs

Cf. A187797, A278700, A352240, A352248.

A279315 Count the primes 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, 4, 2, 1, 0, 6, 0, 1, 12, 1, 0, 12, 0, 1, 6, 1, 0, 2, 0, 1, 0, 0, 2, 0, 0, 1, 30, 1, 0, 0, 2, 0, 0, 1, 0, 2, 0, 1, 12, 1, 0, 2, 0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 1, 6, 1, 0, 0, 2, 0, 0, 1, 0, 2, 0, 1, 2, 1, 0, 0, 2, 0, 0, 1, 0, 6, 0, 1, 0, 0, 2, 0
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 13 2016

Keywords

Links

Crossrefs

Cf. A000720, A010051, A278700, A279481, A279536.

Programs

  • Maple
    with(numtheory): A279315:=n->add( (pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)) * (pi(2*n-i)-pi(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(A279315(n), n=1..100);
  • Mathematica
    f[n_] := Sum[ Boole[PrimeQ[i]] Boole[PrimeQ[ 2n -i]] (PrimePi[ 2n -i] - PrimePi[i -1]) Product[(1 - Abs[Boole[PrimeQ[k]] - Boole[PrimeQ[ 2n -k]]]), {k, i, n}], {i, 3, n}]; Array[f, 80] (* Robert G. Wilson v, Dec 15 2016 *)

Formula

a(n) = Sum_{i=3..n} c(i) * c(2*n-i) * (pi(2*n-i)-pi(i-1)) * (Product_{k=i..n} (1-abs(c(k)-c(2*n-k)))), where pi is the prime counting function (A000720), and c is the prime characteristic (A010051).
From Wesley Ivan Hurt, Dec 17 2016: (Start)
a(n) = A010051(n)*A278700(n)^2+(1-A010051(n))*A278700(n)*(A278700(n)+1).
a(n) <= A279536(n). (End)

A279729 Sum of all the 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, 6, 8, 20, 12, 14, 0, 36, 0, 22, 72, 26, 0, 90, 0, 34, 72, 38, 0, 42, 0, 46, 0, 0, 52, 0, 0, 58, 300, 62, 0, 0, 68, 0, 0, 74, 0, 78, 0, 82, 252, 86, 0, 90, 0, 94, 0, 0, 100, 0, 0, 212, 0, 0, 112, 0, 0, 118, 240, 122, 0, 0, 128, 0, 0, 134, 0, 138, 0, 142, 144, 146, 0
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 17 2016

Keywords

Links

Crossrefs

Cf. A278700, A279727, A279728.

Programs

  • Maple
    with(numtheory): A279729:=n->2*n*add((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(A279729(n), n=1..100);
  • Mathematica
    f[n_, x_: 0] := Sum[(If[x == 0, i, 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}]; Table[f@ n + f[n, 1], {n, 100}] (* Michael De Vlieger, Dec 18 2016 *)

Formula

a(n) = 2n * A278700(n).
a(n) = A279727(n) + A279728(n).

A279481 Count the primes appearing in each interval [p,q] where (p,q) is a Goldbach partition of 2n, and then add the results.

Original entry on oeis.org

0, 0, 1, 2, 4, 2, 5, 8, 6, 9, 13, 12, 14, 10, 12, 12, 24, 22, 9, 20, 24, 27, 29, 38, 36, 24, 39, 29, 33, 43, 24, 58, 58, 17, 52, 60, 53, 63, 80, 46, 54, 87, 70, 46, 100, 62, 58, 87, 31, 79, 104, 71, 87, 119, 99, 116, 152, 114, 94, 181, 54, 82, 144, 39, 116, 133
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 12 2016

Keywords

Links

Crossrefs

Cf. A000720, A010051, A045917, A278700, A279103.

Programs

  • Maple
    with(numtheory): A279481:=n->add( (pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)) * (pi(2*n-i)-pi(i-1)), i=2..n): 0,0,seq(A279481(n), n=3..100);
  • Mathematica
    f[n_] := Sum[ Boole[ PrimeQ[ i]] Boole[ PrimeQ[ 2n -i]] (PrimePi[ 2n -i] - PrimePi[i -1]), {i, 2, n}]; f[2] = 0; Array[ f, 80] (* Robert G. Wilson v, Dec 15 2016 *)

Formula

a(n) = Sum_{i=2..n} A010051(i)*A010051(2*n-i)*(pi(2*n-i)-pi(i-1)) for n > 2.

A352297 Even numbers with exactly 1 pair of Goldbach partitions, (p,q) and (r,s) with p,q,r,s prime and p < r <= s < q, such that all integers in the open intervals (p,r) and (s,q) are composite.

Original entry on oeis.org

10, 16, 18, 22, 34, 42, 46, 64, 82, 96, 98, 110, 136, 140, 154, 160, 188, 190, 194, 218, 224, 230, 236, 244, 256, 274, 280, 308, 314, 338, 340, 350, 368, 370, 382, 388, 394, 398, 400, 404, 422, 428, 440, 446, 452, 466, 470, 488, 494, 500, 512, 514, 524, 536, 574, 578, 580, 586
Offset: 1

Views

Author

Wesley Ivan Hurt, Mar 11 2022

Keywords

Examples

			82 is in the sequence since it has exactly one pair of Goldbach partitions, namely (23,59) and (29,53), such that all integers in the open intervals (23,29) and (53,59) are composite.

Links

Crossrefs

Cf. A187797, A278700, A352248, A352283, A352240.
Cf. See A352351, A352352, A352353, and A352354 for values of the corresponding primes p, q, r, and s.

Formula

a(n) = A352351(n) + A352352(n) = A352353(n) + A352354(n).

A352442 Largest prime "r" among all pairs of Goldbach partitions of A352240(n), (p,q) and (r,s) with p,q,r,s prime and p < r <= s < q, such that all integers in the open intervals (p,r) and (s,q) are composite.

Original entry on oeis.org

5, 5, 7, 5, 11, 13, 5, 17, 13, 5, 19, 17, 29, 5, 23, 31, 29, 19, 29, 41, 37, 17, 37, 43, 53, 41, 37, 29, 53, 59, 53, 61, 53, 41, 67, 59, 47, 71, 5, 47, 29, 79, 71, 73, 83, 53, 83, 37, 59, 83, 37, 29, 71, 29, 101, 103, 107, 67, 89, 73, 67, 59, 101, 79, 59, 107, 79, 113, 5, 109
Offset: 1

Views

Author

Wesley Ivan Hurt, Mar 16 2022

Keywords

Comments

See A352240.

Examples

			a(12) = 17; A352240(12) = 54 has 3 pairs of Goldbach partitions (7,47),(11,43); (11,43),(13,41); and (13,41),(17,37); with all integers composite in the open intervals (7,11) and (43,47), (11,13) and (41,43), and, (13,17) and (37,41) respectively. The largest prime "r" among the Goldbach pairs is 17.

Links

Crossrefs

Cf. A187797, A278700, A352240, A352248, A352283.
Cf. A352443, A352444, A352445.

Formula

a(n) = A352240(n) - A352443(n).

A352443 Smallest prime "s" among all pairs of Goldbach partitions of A352240(n), (p,q) and (r,s) with p,q,r,s prime and p < r <= s < q, such that all integers in the open intervals (p,r) and (s,q) are composite.

Original entry on oeis.org

5, 11, 11, 17, 13, 17, 29, 19, 29, 41, 29, 37, 31, 59, 43, 41, 47, 59, 53, 43, 53, 79, 61, 59, 53, 67, 73, 83, 61, 61, 73, 71, 83, 97, 73, 83, 97, 79, 149, 109, 131, 83, 97, 101, 97, 131, 103, 151, 131, 109, 157, 167, 127, 173, 103, 107, 109, 151, 131, 149, 157, 167, 127
Offset: 1

Views

Author

Wesley Ivan Hurt, Mar 16 2022

Keywords

Comments

See A352240.

Examples

			a(12) = 37; A352240(12) = 54 has 3 pairs of Goldbach partitions (7,47),(11,43); (11,43),(13,41); and (13,41),(17,37); with all integers composite in the open intervals (7,11) and (43,47), (11,13) and (41,43), and, (13,17) and (37,41) respectively. The smallest prime "s" among all Goldbach pairs is 37.

Links

Crossrefs

Cf. A187797, A278700, A352240, A352248, A352283.
Cf. A352442, A352444, A352445.

Formula

a(n) = A352240(n) - A352442(n).

A352444 Largest prime "q" among all pairs of Goldbach partitions of A352240(n), (p,q) and (r,s) with p,q,r,s prime and p < r <= s < q, such that all integers in the open intervals (p,r) and (s,q) are composite.

Original entry on oeis.org

7, 13, 13, 19, 19, 23, 31, 31, 31, 43, 43, 47, 47, 61, 61, 61, 73, 73, 59, 73, 83, 83, 67, 83, 103, 103, 79, 109, 109, 113, 113, 113, 89, 131, 79, 139, 139, 139, 151, 151, 137, 151, 151, 167, 167, 181, 181, 157, 137, 181, 163, 193, 193, 199, 199, 199, 199, 157, 173, 193, 163
Offset: 1

Views

Author

Wesley Ivan Hurt, Mar 16 2022

Keywords

Comments

See A352240.

Examples

			a(12) = 47; A352240(12) = 54 has 3 pairs of Goldbach partitions (7,47),(11,43); (11,43),(13,41); and (13,41),(17,37); with all integers composite in the open intervals (7,11) and (43,47), (11,13) and (41,43), and, (13,17) and (37,41) respectively. The largest prime "q" among all Goldbach pairs is 47.

Links

Crossrefs

Cf. A187797, A278700, A352240, A352248, A352283.
Cf. A352442, A352443, A352445.

Formula

a(n) = A352240(n) - A352445(n).
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