Marcin Barylski - cp's OEIS frontend

A374613 Prime numbers p with the property that neither of 2*(p+next prime to p)+/-1 is prime.

29, 71, 73, 79, 83, 127, 131, 137, 167, 173, 193, 197, 211, 223, 229, 233, 239, 251, 269, 331, 347, 349, 373, 419, 421, 431, 433, 439, 457, 467, 487, 503, 509, 541, 563, 577, 587, 619, 641, 647, 653, 659, 661, 691, 719, 727, 733, 743, 751, 797, 809, 811, 821, 829, 839, 853, 859, 863, 887

Offset: 1

Marcin Barylski, Jul 14 2024

nonn

Conjecture: this sequence is infinite.

			2 is not a term because 2*(2+3)+1=11 is a prime.
5 is not a term because 2*(5+7)-1=23 is a prime.
29 is a term because neither 2(29+31)+1 nor 2(29+31)-1 is a prime.

Prime[Select[Range[155], !PrimeQ[2(Prime[#]+Prime[#+1])-1] && !PrimeQ[2(Prime[#]+Prime[#+1])+1]&]] (* Stefano Spezia, Jul 15 2024 *)

A346659 Primes that are not of the form p*q +- 2 where p and q are primes (not necessarily distinct).

Original entry on oeis.org

3, 5, 29, 43, 61, 73, 101, 103, 107, 137, 149, 151, 173, 191, 193, 197, 227, 229, 241, 271, 277, 281, 283, 313, 347, 349, 421, 431, 433, 457, 461, 463, 523, 569, 601, 607, 617, 619, 641, 643, 659, 661, 727, 821, 823, 827, 857, 859, 883, 929, 1019, 1021, 1031

Offset: 1

Marcin Barylski, Jul 27 2021

nonn

Conjecture: this sequence is infinite.

			2 is not a term because 2 = 2*2 - 2.
3 is a term because neither 1 (3-2) nor 5 (3+2) is a product of two primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A207526 (complementary sequence).

q:= n-> andmap(x-> numtheory[bigomega](x)<>2, [n-2, n+2]):
select(q, [ithprime(i)$i=1..200])[];  # Alois P. Heinz, Jul 30 2021

Select[Range[3, 1000], PrimeQ[#] && PrimeOmega[# - 2] != 2 && PrimeOmega[# + 2] != 2 &] (* Amiram Eldar, Jul 29 2021 *)

from sympy import factorint, primerange
def semiprime(n): return sum(e for e in factorint(n).values()) == 2
def ok(p): return not semiprime(p-2) and not semiprime(p+2)
def aupto(limit): return list(filter(ok, primerange(1, limit+1)))
print(aupto(1031)) # Michael S. Branicky, Jul 29 2021

More terms from Michael S. Branicky, Jul 29 2021

A333779 Positive numbers m used to build entire prime set by m +/- n without duplication or 0 if there is no such m.

Original entry on oeis.org

2, 4, 9, 16, 27, 42, 23, 60, 51, 70, 93, 120, 85, 114, 153, 56, 165, 174, 155, 132, 213, 218, 201, 234, 253, 288, 225, 254, 135, 360, 323, 342, 315, 274, 303, 384, 395, 420, 405, 440, 357, 420, 481, 534, 465, 454

Offset: 0

Marcin Barylski, Apr 05 2020

nonn

Conjecture: every prime is eventually constructed by the sequence.
Taking into account first 10 terms: a(0),a(1),...a(9) = [2, 4, 9, 16, 27, 42, 23, 60, 51, 70] it is possible to build the following primes: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 79], the only not covered (yet) primes  <= 79 are: [41, 71, 73]. 73 will be covered by a(12)=85 (73=85-12), and both 41 and 71 by a(15)=56 (41=56-15, 71=56+15).
The truth of Polignac's conjecture would imply that all terms are well defined. - Rémy Sigrist, Apr 26 2020
a(n) > 0 for 1 <= n <= 10^6. - David A. Corneth, Jun 06 2020

			a(0)=2, because 2=2+0=2-0 and 2 is prime.
a(1)=4, because 3=4-1, 5=4+1, both 3 and 5 are primes, not covered yet.
a(1) is not 3 because 3+1=4 is not a prime number.
a(2)=9, because 7=9-2, 11=9+2, both 7 and 11 are primes, not covered yet.
a(2) is not 5 (although 5-2=3 and 5+2=7, both are primes) because 3 is already covered by a term a(1) - this sequence is without duplication.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Marcin Barylski, C++ program for generating A333779
Marcin Barylski, On the symmetry of primes

Cf. A000040, A000230, A020483.

Nest[Function[{t, i}, Append[t, Block[{k = 2, s}, While[! AllTrue[Set[s, k + i {-1, 1}], And[PrimeQ@ #, FreeQ[t[[All, -1]], #] ] &], k++]; {k, s}] ]] @@ {#, Length@ #} &, {{2, {2}}}, 60][[All, 1]] (* Michael De Vlieger, May 03 2020 *)

{ p=2; pp=[]; for (n=0,  45, for (k=1, oo, while (#pppp[#pp], pp = concat(pp, p); p = nextprime(p+1);); if (setsearch(pp, pp[k]+2*n), print1 (pp[k]+n", "); pp = setminus(pp, Set([pp[k], pp[k]+2*n])); break))) } \\ Rémy Sigrist, Jun 06 2020

More terms from Michael De Vlieger, May 03 2020

A333122 Numbers m such that m = prime(k) + prime(k+5) = prime(k+1) + prime(k+4) for some k.

Original entry on oeis.org

24, 30, 60, 84, 102, 210, 234, 288, 330, 378, 420, 426, 496, 528, 588, 594, 624, 690, 1050, 1156, 1200, 1218, 1302, 1336, 1410, 1470, 1484, 1638, 1650, 1680, 1686, 1716, 1734, 1740, 1746, 1788, 1848, 1908, 1918, 1930, 2052, 2154, 2226, 2364, 2410, 2580, 2892, 2934, 3168, 3524, 4080

Offset: 1

Marcin Barylski, Mar 08 2020

nonn

Terms are always even because all primes used in this sequence are odd.

Conjecture: this sequence is infinite.

			a(1)=24 because prime(3)+prime(8)=prime(4)+prime(7)=5+19=7+17.

Cf. A022889 (the prime(k) primes), A105093 (similar sequence).

(#[[1]] + #[[6]]) & /@ Select[ Partition[ Prime@ Range@ 320, 6, 1], #[[1]] + #[[6]] == #[[2]] + #[[5]] &] (* Giovanni Resta, Mar 08 2020 *)

from sympy import nextprime
A333122_list, plist = [], [2,3,5,7,11,13]
while len(A333122_list) < 10000:
    m = plist[0]+plist[5]
    if m == plist[1]+plist[4]:
        A333122_list.append(m)
    plist = plist[1:] + [nextprime(plist[-1])] # Chai Wah Wu, Mar 30 2020

A328179 Number of distinct primes required to satisfy the Strong Goldbach Conjecture for all even numbers <= 2n.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17

Offset: 1

Marcin Barylski, Oct 06 2019

nonn

The Strong Goldbach Conjecture asserts that all positive even integers >=4 can be expressed as the sum of two primes.

If the Strong Goldbach Conjecture is true, then a(n) > 0 for all n > 1 and a(n) <= a(n+1) for all n.

			a(1)=0 because 2 does not have any Goldbach partition.
a(2)=1 because 4=2+2 and 2 is the only prime required for all even numbers <= 4.
a(3)=2 because 4=2+2 and 6=3+3, thus 2 and 3 are required for expressing all even numbers <= 6.
a(7)=4 because using {2,3,5,7} it is possible to build all even numbers <= 14.
a(8)=5 because using either {2,3,5,7,11} or {2,3,5,7,13} it is possible to build all even numbers <= 16.
a(10)=5 because {2,3,5,7,13} are enough to build all even numbers <= 20.

Marcin Barylski, Comparison of A328179 with pi(n) for the first 10^4 numbers
Marcin Barylski, Redundant Primes In Goldbach Partitions

A321221 Numbers of the form 6n-2 which are not a sum of two numbers that are the lesser of twin primes.

Original entry on oeis.org

4, 94, 400, 514, 784, 904, 1114, 1144, 1264, 1354, 3244, 4204

Offset: 1

Marcin Barylski, Oct 31 2018

Conjecture: This sequence is finite.

If this sequence is finite, then the Goldbach Strong Conjecture is true. If p1 and p2 are the lesser of twin primes, then q1=p1+2 and q2=p2+2 are also primes (they are the greater of twin primes). If 6n-2 = p1+p2, then 6n = q1+p2 and 6n+2 = q1+q2.

			a(1) = 4 because 4 = 2+2; there are no other Goldbach partitions and 2 is not the lesser of twin primes.
a(2) is not 6 because 6 = 3+3 and 3 is the lesser of twin primes.

Marcin Barylski, On 6k ± 1 Primes in Goldbach Strong Conjecture
Marcin Barylski, C++ program for generating A321221

Cf. A001359, A002375.

aQ[n_]:= (k=1; kmax=(n+2)/6; While[k<=kmax && !AllTrue[{6k-1,6k+1,6(kmax-k)-1,6(kmax-k)+1}, PrimeQ], k++]; k>kmax); Select[6*Range[0,10000]+4,aQ] (* Amiram Eldar, Nov 10 2018 *)

ok(n)={if(n%6 == 4, forstep(k=3, n\2, 2, if(isprime(k) && isprime(k+2) && isprime(n-k) && isprime(n-k+2), return(0))); 1, 0)} \\ Andrew Howroyd, Nov 01 2018

A303603 a(n) is the maximum distance between primes in Goldbach partitions of 2n, or 2n if there are no Goldbach partitions of 2n.

Original entry on oeis.org

0, 0, 0, 2, 4, 2, 8, 10, 8, 14, 16, 14, 20, 18, 16, 26, 28, 26, 24, 34, 32, 38, 40, 38, 44, 42, 40, 50, 48, 46, 56, 58, 56, 54, 64, 62, 68, 70, 68, 66, 76, 74, 80, 78, 76, 86, 84, 82, 60, 94, 92, 98, 100, 98, 104, 106, 104, 110, 108, 106, 96, 102, 100, 90, 124, 122, 128, 126, 124, 134, 136, 134, 132

Offset: 1

Marcin Barylski, Apr 26 2018

nonn

The Goldbach Strong Conjecture is true if and only if a(n) = 2n for some n.
Terms are always even numbers because primes present in Goldbach partitions of n > 4 are odd and n = 4 has just one partition (2+2) where difference is 0.
Conjecture: Only first terms are 0 and all further terms are bigger than 0. Excluding a(1), a(n) = 0 iff the only Goldbach partition of 2n is n+n.

			a(1) = 0 for coherence with other related sequences.
a(2) = 0 because 2 * 2 = 4 = 2 + 2 and max_diff = 2 - 2 = 0.
a(8) = 10 because 2 * 8 = 16 = 5 + 11 = 3 + 13 and max_diff = 13 - 3 = 10.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Marcin Barylski, C++ program
Marcin Barylski, Maximum distance for even numbers < 10^6
Eric Weisstein's MathWorld, Goldbach Partition

Cf. A002372, A002375, A047949, A066285 (minimum distance), A305883.

a[1]=a[2]=0;
a[n_]:=Module[{p=3},While[PrimeQ[2*n-p]!=True,p=NextPrime[p]];2*(n-p)];
a/@Range[73] (* Ivan N. Ianakiev, Jun 27 2018 *)

a(n) = if (n==1, 0, forprime(p=2, , if (isprime(2*n-p), return (2*n-2*p)))); \\ Michel Marcus, Jul 02 2018

a(n) = 2 * A047949(n) if A047949(n) > 0 for n >= 2; a(n) = 2n if A047949(n) = -1. - Alois P. Heinz, Jun 01 2020

A301776 Prime numbers p with the property that all even numbers n (2 < n <= 2p) are the sum of two primes <= p.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 109

Offset: 1

Marcin Barylski, Mar 26 2018

Conjecture: this sequence is finite - it has 7 terms only. Conjecture verified up to first 10^5 primes.

This sequence is related to the Goldbach Strong Conjecture.

			a(1)=2 because all even numbers 2 < n <= 2*2 (there is just one such number: 4) can be expressed as a sum of 2 only: 4=2+2.
a(2)=3 because 4=2+2, 6=3+3.
a(3)=5 because 4=2+2, 6=3+3, 8=5+3, 10=5+5.
a(4)=7 because 4=2+2, 6=3+3, 8=5+3, 10=5+5, 12=5+7, 14=7+7.
a(5)=13 (and is not 11) because 20 cannot be expressed as a sum of two primes from a set {2,3,5,7,11} but all even numbers 2 < n <= 26 can be expressed as a sum of two primes from a set {2,3,5,7,11,13}.

Marcin Barylski, C++ program
Marcin Barylski, Sum building from first primes vs. theoretical maximum - the first 200 rounds of the algorithm. If red and green lines ever touch each other, we have a new term.
Marcin Barylski, Goldbach Strong Conjecture Verification Using Prime Numbers

Cf. A002372 (number of ordered Goldbach partitions).

Select[Prime@ Range[500], Function[p, SameQ[Select[Union@ Map[Total, Tuples[Prime@ Range@ PrimePi@ p, 2]], And[EvenQ@ #, # > p] &], Range[p + 1 + Boole@ EvenQ@ p, 2 p, 2]]]] (* Michael De Vlieger, Apr 10 2018 *)

isok(p) = {vp = primes(primepi(p)); slist = List(); for (i=1, #vp, for (j=1, i, if (!((vp[i]+vp[j]) % 2), listput(slist, vp[i]+vp[j])););); #Set(slist) == (p-1);}
lista(nn) = forprime(p=2, nn, if (isok(p), print1(p, ", "))); \\ Michel Marcus, Apr 09 2018

A295424 Number of distinct twin primes which are in Goldbach partitions of 2n.

Original entry on oeis.org

0, 0, 1, 2, 3, 2, 3, 4, 4, 4, 5, 6, 4, 3, 5, 4, 6, 7, 3, 4, 6, 5, 6, 9, 6, 4, 7, 4, 5, 8, 5, 7, 8, 3, 6, 10, 7, 7, 11, 6, 6, 10, 6, 6, 11, 6, 4, 7, 3, 7, 11, 7, 6, 10, 8, 10, 15, 8, 8, 14, 6, 6, 10, 4, 8, 12, 6, 3, 10, 9, 10, 15, 7, 7, 12, 7, 10, 14, 6, 9, 13, 5, 7

Offset: 1

Marcin Barylski, Feb 12 2018

nonn

Tomas Oliveira e Silva in 2012 experimentally confirmed that all even numbers 4 <= n <= 4 * 10^18 have at least one Goldbach partition (GP) with a prime 9781 or less. Detailed examination of all even numbers less than 10^6 showed that the most popular prime in all GPs is 3 (78497 occurrences), then 5 (70328), then 7 (62185), then 11 (48582), then 13 (40916), then 17 (31091), then 19 (29791) -- all these primes are twin primes. These results gave rise to a hypothesis that twin primes should be rather frequent in GP, especially those relatively small.
Conjecture. Further empirical examinations lead to a hypothesis that all even numbers n > 4 have at least 1 twin prime in GP(n).
a(n) <= A294185(n) + A294186(n).

			a(5) = 3 because 5 * 2 = 10 has 2 ordered Goldbach partitions: 3 + 7 and 5 + 5 and primes 3, 5, 7 are distinct twin primes in this set.

Marcin Barylski, Plot of first 20000 elements of the A295424
Marcin Barylski, C++ program for generating A295424
Marcin Barylski, Studies on Twin Primes in Goldbach Partitions of Even Numbers
Tomas Oliveira e Silva, Goldbach conjecture verification

Cf. A294186, A294185, A001097.

istwin(p) = isprime(p) && (isprime(p-2) || isprime(p+2));
a(n) = {vtp = []; forprime(p= 2, n, if (isprime(2*n-p), if (istwin(p), vtp = concat(vtp, p)); if (istwin(2*n-p), vtp = concat(vtp, 2*n-p)););); #Set(vtp);} \\ Michel Marcus, Mar 01 2018

A294186 Number of distinct greater twin primes which are in Goldbach partitions of 2n.

Original entry on oeis.org

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

Offset: 1

Marcin Barylski, Feb 11 2018

nonn

Tomas Oliveira e Silva in 2012 experimentally confirmed that all even numbers <= 4*10^18 have at least one Goldbach partition (GP) with a prime 9781 or less. Detailed examination of all even numbers < 10^6 showed that the most popular prime in all GPs is 3 (78497 occurrences), then 5 (70328), then 7 (62185), then 11 (48582), then 13 (40916), then 17 (31091), then 19 (29791) - all these primes are twin primes. These results gave rise to a hypothesis that twin primes should be rather frequent in GP, especially those relatively small.
Further empirical experiments demonstrated, surprisingly, there are in general two categories of even numbers n: category 1 - with 0, 1, or 2 distinct greater twin primes in all GPs(n), and category 2 - with fast increasing number of distinct greater twin primes in GPs(n).
Is a(n) = A294185(n-1)? - R. J. Mathar, Mar 22 2024

			a(5)=2 because 2*5=10 has two ordered Goldbach partitions: 3+7 and 5+5. 5 is a greater twin prime (because 3 and 5 are twin primes), 7 is a greater twin prime (because 5 and 7 are twin primes).

Marcin Barylski, Plot of first 20000 elements of the A294186
Marcin Barylski, C++ program for generating A294186
Tomas Oliveira e Silva, Goldbach conjecture verification

Cf. A002372 (number of ordered Goldbach partitions), A006512 (greater of twin primes), A294185, A295424.

isgtwin(p) = isprime(p) && isprime(p-2);
a(n) = {vtp = []; forprime(p = 2, n, if (isprime(2*n-p), if (isgtwin(p), vtp = concat(vtp, p)); if (isgtwin(2*n-p), vtp = concat(vtp, 2*n-p)););); #Set(vtp);} \\ Michel Marcus, Mar 01 2018

cp's OEIS Frontend

User: Marcin Barylski

A374613 Prime numbers p with the property that neither of 2*(p+next prime to p)+/-1 is prime.

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A346659 Primes that are not of the form p*q +- 2 where p and q are primes (not necessarily distinct).

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A333779 Positive numbers m used to build entire prime set by m +/- n without duplication or 0 if there is no such m.

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A333122 Numbers m such that m = prime(k) + prime(k+5) = prime(k+1) + prime(k+4) for some k.

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A328179 Number of distinct primes required to satisfy the Strong Goldbach Conjecture for all even numbers <= 2n.

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A321221 Numbers of the form 6n-2 which are not a sum of two numbers that are the lesser of twin primes.

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A303603 a(n) is the maximum distance between primes in Goldbach partitions of 2n, or 2n if there are no Goldbach partitions of 2n.

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A301776 Prime numbers p with the property that all even numbers n (2 < n <= 2p) are the sum of two primes <= p.

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