A278700 -id:A278700 - cp's OEIS frontend

A352445 Smallest prime "p" 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.

3, 3, 5, 3, 5, 7, 3, 5, 11, 3, 5, 7, 13, 3, 5, 11, 3, 5, 23, 11, 7, 13, 31, 19, 3, 5, 31, 3, 5, 7, 13, 19, 47, 7, 61, 3, 5, 11, 3, 5, 23, 11, 17, 7, 13, 3, 5, 31, 53, 11, 31, 3, 5, 3, 5, 11, 17, 61, 47, 29, 61, 47, 29, 73, 3, 5, 73, 7, 3, 5, 11, 83, 17, 23, 37, 29, 3, 5, 23

Offset: 1

Wesley Ivan Hurt, Mar 16 2022

nonn

			a(12) = 7; 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 "p" among all Goldbach pairs is 7.

Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A187797, A278700, A352240, A352248, A352283.

Cf. A352442, A352443, A352444.

a(n) = A352240(n) - A352444(n).

A279103 Number of Goldbach partitions (p,q) of 2n such that there exists a prime r in p < r < q that does not appear as a part in any Goldbach partition of p+q = 2n.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, Dec 06 2016

Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A002375, A010051, A278700.

a(n) = A002375(n) - A278700(n).

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

A279730 Partial sums of A279729.

Original entry on oeis.org

0, 0, 6, 14, 34, 46, 60, 60, 96, 96, 118, 190, 216, 216, 306, 306, 340, 412, 450, 450, 492, 492, 538, 538, 538, 590, 590, 590, 648, 948, 1010, 1010, 1010, 1078, 1078, 1078, 1152, 1152, 1230, 1230, 1312, 1564, 1650, 1650, 1740, 1740, 1834, 1834, 1834, 1934

Offset: 1

Wesley Ivan Hurt, Dec 17 2016

Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A278700, A279727, A279728, A279729.

with(numtheory): A279730:=n->2*add(add(j * (pi(i)-pi(i-1)) * (pi(2*j-i)-pi(2*j-i-1)) * (product(1-abs((pi(k)-pi(k-1))-(pi(2*j-k)-pi(2*j-k-1))), k=i..j)), i=3..j), j=1..n): seq(A279730(n), n=1..40);

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}]; Accumulate@ Table[f@ n + f[n, 1], {n, 50}] (* Michael De Vlieger, Dec 18 2016 *)

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

Wesley Ivan Hurt, Oct 19 2017

nonn

			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.

Bert Dobbelaere, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

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

More terms from Bert Dobbelaere, Sep 15 2019

cp's OEIS Frontend

A352445 Smallest prime "p" 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.

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A279103 Number of Goldbach partitions (p,q) of 2n such that there exists a prime r in p < r < q that does not appear as a part in any Goldbach partition of p+q = 2n.

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A279730 Partial sums of A279729.

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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.

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