A093161 -id:A093161 - cp's OEIS frontend

A279040 Even numbers 2k such that the smallest prime p satisfying p+q=2k (q prime) is greater than or equal to sqrt(k).

4, 6, 8, 10, 12, 14, 16, 18, 24, 28, 30, 36, 38, 42, 48, 54, 60, 68, 80, 90, 96, 98, 122, 124, 126, 128, 148, 150, 190, 192, 208, 210, 212, 220, 222, 224, 302, 306, 308, 326, 330, 332, 346, 368, 398, 418, 458, 488, 518, 538, 540, 542, 556, 640, 692, 710, 796, 854, 908, 962, 968, 992, 1006

Offset: 1

Corinna Regina Böger, Dec 04 2016

nonn

a(n) is an extension of A244408.
It is conjectured that a(230) = 503222 is the last term. Oliveira e Silva's work shows that there are no more terms below 4*10^18.
The sequence definition is equivalent to: "Even integers k such that there exists a prime p with p = min{q: q prime and (k - q) prime} and k < 2*p^2" and therefore this is a member of the EGN- family (Cf. A307782). - Corinna Regina Böger, May 01 2019

			The smallest prime for 42 is 5 with 5 > sqrt(21), but not smaller than sqrt(42), and therefore 42 does not belong to A244408. The smallest prime for 38 is 7, and 7 >= sqrt(38), and therefore 38 also belongs to A244408.

Corinna Regina Böger, Table of n, a(n) for n = 1..230
Tomás Oliveira e Silva, Goldbach conjecture verification
Index entries for sequences related to Goldbach conjecture

Cf. A002373, A020481, A025018, A025019, A093161, A244408, A307542, A307782

Select[Range[4, 1006, 2], Function[n, Select[#, PrimeQ@ Last@ # &][[1, 1]] >= Sqrt[n/2] &@ Map[{#, n - #} &, Prime@ Range@ PrimePi@ n]]] (* Michael De Vlieger, Dec 06 2016 *)

isok(n) = forprime(p=2, n, if (isprime(n-p), if (p >= sqrt(n/2), return(1), return(0))));
lista(nn) = forstep(n=2, nn, 2, if (isok(n), print1(n, ", "))) \\ Michel Marcus, Dec 04 2016

A307542 Even integers k such that there exists a prime p with p = min{q: q prime and (k - q) prime} and (k - p) < p^2.

Original entry on oeis.org

4, 6, 8, 10, 12, 18, 24, 28, 30, 38, 54, 98, 122, 124, 126, 128, 220, 302, 308, 332, 346, 368, 488, 556, 854, 908, 962, 968, 992, 1144, 1150, 1274, 1354, 1360, 1362, 1382, 1408, 1424, 1532, 1768, 1856, 1928, 2078, 2188, 2200, 2438, 2512, 2530, 2618, 2642, 3458, 3526, 3818, 3848

Offset: 1

Corinna Regina Böger, Apr 14 2019

nonn

This sequence is an extension of A244408. It is equivalent to "Even numbers 2n such that the smallest prime p satisfying p+q=2n (p, q prime, p<=q) also satisfies p^2+p>2n." If p satisfies additionally p^2 < 2n the corresponding even numbers do not belong to A244408. These numbers are 10, 28, 54, 124, 368, 968, 3526. It is conjectured that a(81)=63274 is the last term. There are no more terms below 4*10^18.

			10 = 3 + 7, 3^2 = 9 < 10 and 9 > 7 = q, therefore it is in this sequence.

Corinna Regina Böger, Table of n, a(n) for n = 1..81

Cf. A244408, A093161, A279040.

isS := proc(n) local p; for p from 2 while p^2 < (n-p) do
if isprime(p) and isprime(n-p) then return false fi od; true end:
isa := n -> irem(n, 2) = 0 and isS(n): select(isa, [$4..3848]); # Peter Luschny, Apr 26 2019

Select[Range[4, 4000, 2], #2 > Sqrt@ #1 & @@ SelectFirst[IntegerPartitions[#, {2}], AllTrue[#, PrimeQ] &] &] (* Michael De Vlieger, Apr 21 2019 *)

noSpecialGoldbach(n) = forprime(p=2, n/2, if(p^2+p2 && n%2 == 0 && noSpecialGoldbach(n)

A307782 Even integers k such that there exists a prime p with p=min{q: q prime and (k-q) prime} and k < p^3.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 36, 38, 42, 48, 52, 54, 58, 60, 66, 68, 72, 78, 80, 84, 88, 90, 94, 96, 98, 102, 108, 114, 118, 120, 122, 124, 126, 128, 138, 146, 148, 150, 158, 164, 174, 180, 188, 190, 192, 206, 208, 210, 212, 218, 220, 222, 224, 240, 248, 250, 252, 258, 264, 270, 278, 290, 292, 294, 300, 302, 304, 306, 308, 324, 326, 328, 330, 332, 338, 346

Offset: 1

Corinna Regina Böger, Apr 28 2019

nonn

[Please keep the larger data section as it shows where the sequence first differs from A093161.]
This is another member of a family of sequences concerning the Strong Goldbach Conjecture, which I define as follows: Let (x, y, z) be real numbers with x >= 2, y > 0, z >= 0. An even integer k is then called an (x, y, z) Extraordinary Goldbach Number (EGN) if there exists a prime p with p=min{q: q prime and (k-q) prime} and (k - z*p) < y*p^x. a(n) represents the (3, 1, 0) extraordinary Goldbach numbers. A093161 consists of (3, 1, 1) EGN, A307542 are the (2, 1, 1) EGN, A279040 are the (2, 2, 0) EGN and A244408 are the (2, 1, 0).
a(104809) is very probably the last term and there are no more terms below 4*10^18.
There are only 11 terms in A093161 that are not in this sequence; these are 344, 1338, 12184, 12186, 24400, 148912, 1030342, 2571406, 3308008, 5929868, 15813352.

			344 is not in the sequence, because the smallest prime p for 344 is 7 with 7^3 = 343 < 344, whereas it is in A093161 due to 344 - 7 = 337 < 7^3.

Corinna Regina Böger, Table of n, a(n) for n = 1..10000
Corinna Regina Böger, a-file, Table of n, a(n) for n = 1..104809

Cf. A093161, A244408, A279040, A307542.

extraordinaryGoldbach(x,y,z,k) = forprime(p=2, k/2, if(isprime(k-p), if(y*p^x+z*p>=k, return(1),return(0)))); 0
is(n) = n%2 == 0 && extraordinaryGoldbach(3, 1, 0, n)

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A279040 Even numbers 2k such that the smallest prime p satisfying p+q=2k (q prime) is greater than or equal to sqrt(k).

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A307542 Even integers k such that there exists a prime p with p = min{q: q prime and (k - q) prime} and (k - p) < p^2.

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A307782 Even integers k such that there exists a prime p with p=min{q: q prime and (k-q) prime} and k < p^3.

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