Corinna Regina Böger - cp's OEIS frontend

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.

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)

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)

A274189 Even numbers 2n that satisfy an extended Goldbach conjecture: They have at least one Goldbach partition 2n = p + q (p <= q; p, q prime) that satisfies p <= sqrt(n), at least one with sqrt(n) < p <= sqrt(2n) and at least one with p > sqrt(2n).

Original entry on oeis.org

34, 46, 50, 66, 74, 78, 86, 138, 142, 160, 162, 168, 170, 174, 176, 178, 180, 184, 186, 194, 202, 204, 206, 234, 236, 238, 240, 242, 246, 252, 254, 264, 270, 276, 282, 284, 290, 294, 296, 298, 300, 310, 320, 324, 328, 334, 348, 354, 364, 366, 370, 372, 376, 378, 384, 386, 390, 392, 396, 400

Offset: 1

Corinna Regina Böger, Dec 11 2016

nonn

This sequence contains all even numbers that are not in A279040 or in A273457. I have verified numerically for all even numbers 4 < 2n < 4*10^10 that a Goldbach partition with the additional condition p > sqrt(2n) exists. It is conjectured that a(n) = 2*(n+12987) for all n > 7838315. If this conjecture is true, all even numbers 2n > 15702604 have all three types of Goldbach partitions and therefore satisfy the "extended Goldbach conjecture".

			a(1) = 34 = 3 + 31 = 5 + 29 = 11 + 23 = 17 + 17. Since 3 < sqrt(17) < 5 < sqrt(34) < 11 < 17, all three types of Goldbach partitions exist for 34.

Corinna Regina Böger, Table of n, a(n) for n = 1..100000
Index entries for sequences related to Goldbach conjecture

Cf. A002373, A020481, A025018, A025019, A244408, A273457, A279040.

GoldbachRange(n,mn,mx)=forprime(p=mn,mx, if(isprime(n-p), return(1))); 0
is(n)=n%2==0 && GoldbachRange(n, 2, sqrtint(n/2)) && GoldbachRange(n, sqrtint(n/2-1)+1, sqrtint(n)) && GoldbachRange(n, sqrtint(n-1)+1, n/2) \\ Charles R Greathouse IV, Dec 16 2016

A273457 Even numbers 2n that do not have a Goldbach partition 2n = p + q (p < q; p, q prime) satisfying sqrt(n) < p <= sqrt(2n).

Original entry on oeis.org

2, 4, 6, 8, 12, 18, 20, 22, 24, 26, 30, 32, 38, 40, 44, 52, 56, 58, 62, 64, 70, 72, 76, 82, 84, 88, 92, 94, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 126, 128, 130, 132, 134, 136, 140, 144, 146, 152, 154, 156, 158, 164, 166, 172, 182, 188, 196, 198, 200, 214

Offset: 1

Corinna Regina Böger, Dec 11 2016

nonn

This is an extension of A244408.
There are 74 elements of A279040 that are also in this sequence. These common elements are in A244408.
It is conjectured that a(12831) = 15702604 is the last term. There are no more terms below 4*10^10.

			32 is in the sequence because 32 has two Goldbach partitions: 32 = 3 + 29 with 3 < sqrt(16) and 32 = 13 + 19 with 13 > sqrt(32).

Corinna Regina Böger, Table of n, a(n) for n = 1..12831

Cf. A002373, A020481, A025018, A025019, A244408, A279040.

noGoldbatSqrQ[n_] := Block[{p = NextPrime[Sqrt[n/2]]}, While[2p < n && !PrimeQ[n - p], p = NextPrime@ p]; p > Sqrt[n]]; noGoldbatSqrQ[4] = True; Select[2Range[107], noGoldbatSqrQ] (* Robert G. Wilson v, Dec 15 2016 *)

noSpecialGoldbach(n) = forprime(p=sqrtint(n/2-1) + 1, sqrtint(n), if(p<(n-p) && isprime(n-p), return(0))); 1
is(n) = n%2 == 0 && noSpecialGoldbach(n)

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

Original entry on oeis.org

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

A268757 Unhappycab numbers: the smallest unhappy number that is the sum of two cubes of unhappy numbers in n different ways.

Original entry on oeis.org

16, 4104, 119824488, 74213505639000, 6123582409620312000

Offset: 1

Corinna Regina Böger, Feb 14 2016

a(6) <= 802780294845778147551744 = 9851898^3+92902278^3 = 27391106^3+92139262^3 = 52144168^3+87109808^3 = 56241696^3+85493352^3 = 58805572^3+84316316^3 = 72408488^3+75075088^3.

			a(1) = 16 = 2^3+2^3;
a(2) = 4104 = 2^3+16^3 = 9^3+15^3;
a(3) = 119824488 = 11^3+493^3 = 90^3+492^3 = 346^3+428^3;
a(4) = 74213505639000 = 5895^3+41985^3 = 20392^3+40358^3 = 20880^3+40230^3 = 32790^3+33900^3;
a(5) = 6123582409620312000 = 193935^3+1828785^3 = 539195^3+1813765^3 = 1026460^3+1714760^3 = 1107120^3+1682940^3 = 1157590^3+1659770^3.

Cf. A011541, A031177, A268583.

a(n) >= A011541(n) for n > 0. It is conjectured that there is no equality.

A268583 Happycab numbers: the smallest happy number that is the sum of two cubes of happy numbers in n different ways.

Original entry on oeis.org

7859, 681179092750, 4466203788801326865111

Offset: 1

Corinna Regina Böger, Feb 07 2016

a(n)>3.6*10^24 for n >=4. This lower bound was established by exhaustive crowd computing.
Upper bounds:
a(4)<= 3915335521240189321820073984
=467175960^3+1562319144^3
=569783808^3+1550898288^3
=1085420968^3+1381483928^3
=1157553216^3+1332193392^3
a(5)<=
1508202165690304620654479410485250200981504
=12156471201588^3+114634141265868^3
=33798481396036^3+113692639229372^3
=64341822928208^3+107486686558048^3
=69397851840576^3+105492106344912^3
=72561474196232^3+104039736026296^3

			a(1) = 7859 = 10^3+19^3;
a(2) = 681179092750 = 4365^3+8425^3 = 5275^3+8115^3;
a(3) = 4466203788801326865111 = 6193863^3+16170804^3 = 8456292^3+15688647^3 = 9457695^3+15354846^3.

Cf. A011541, A007770.

cp's OEIS Frontend

User: Corinna Regina Böger

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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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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A274189 Even numbers 2n that satisfy an extended Goldbach conjecture: They have at least one Goldbach partition 2n = p + q (p <= q; p, q prime) that satisfies p <= sqrt(n), at least one with sqrt(n) < p <= sqrt(2n) and at least one with p > sqrt(2n).

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A273457 Even numbers 2n that do not have a Goldbach partition 2n = p + q (p < q; p, q prime) satisfying sqrt(n) < p <= sqrt(2n).

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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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A268757 Unhappycab numbers: the smallest unhappy number that is the sum of two cubes of unhappy numbers in n different ways.

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A268583 Happycab numbers: the smallest happy number that is the sum of two cubes of happy numbers in n different ways.

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