A065978 -id:A065978 - cp's OEIS frontend

A066285 a(n) is the minimal difference between primes p and q whose sum is 2n.

0, 0, 2, 0, 2, 0, 6, 4, 6, 0, 2, 0, 6, 4, 6, 0, 2, 0, 6, 4, 18, 0, 10, 12, 6, 8, 18, 0, 2, 0, 18, 8, 6, 12, 10, 0, 18, 4, 6, 0, 2, 0, 6, 4, 30, 0, 10, 24, 6, 16, 18, 0, 14, 24, 6, 8, 30, 0, 2, 0, 18, 8, 6, 12, 10, 0, 30, 4, 6, 0, 2, 0, 30, 8, 6, 12, 10, 0, 18, 4, 30, 0, 10, 24, 6, 28, 18, 0

Offset: 2

Dean Hickerson, Dec 12 2001

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 the difference is 0. a(n) = 0 iff n is prime. - Marcin Barylski, Apr 28 2018

Cf. A002372, A047160, A065978, A066286, A303603.

a[n_] := For[p=n, True, p--, If[PrimeQ[p]&&PrimeQ[2n-p], Return[2n-2p]]]

a(n) = {forstep(k=n, 1, -1, if (isprime(k) && isprime(2*n-k), return(2*n-2*k)););} \\ Michel Marcus, Jun 01 2020

a(n) = 2 * A047160(n). - Alois P. Heinz, Jun 01 2020

A107926 The least number k such that there are primes p and q with p - q = 2*n, p + q = k, and p the least such prime >= k/2.

Original entry on oeis.org

4, 8, 18, 16, 54, 48, 50, 108, 102, 44, 234, 444, 98, 228, 174, 92, 414, 432, 242, 516, 582, 256, 1182, 672, 406, 612, 846, 272, 1038, 984, 442, 1776, 1902, 292, 1074, 636, 1054, 3312, 1122, 476, 1398, 1464, 530, 1728, 2730, 572, 2706, 3348, 682, 2844, 3342

Offset: 0

Gilmar J. Rodriguez (Gilmar.Rodriguez(AT)nwfwmd.state.fl.us) and Robert G. Wilson v, Jun 16 2005

nonn

From the Goldbach conjecture.
A107926 = 2*A103147 by definition.
a(3n)> a(3n-2), a(3n-1), a(3n+1) & a(3n+2) for all n > 0 except for n = 1, 2, 12, 19, 20 or 41.
Of those values found so far a(3n+2) > a(3n+1) by ~8%. - Robert G. Wilson v, Nov 03 2013
Except for 1, all indices, i, not congruent to 0 (mod 3), a(i) is congruent to 0 (mod 6) and for all indices, i, congruent to 0 (mod 3), a(i) is not congruent to 0 (mod 6). Of those not congruent to 0 (mod 6), those congruent to 2 outnumber those congruent to 4, about 8 to 7. Robert G. Wilson v, Nov 03 2013

			a(0) = 4 because 4=2+2 and 2-2=0.
a(1) = 8 because 8 is the least number with 8=p+q and p-q=2 for primes p and q.
a(2) = 18 because 18=7+11 and the primes 7 and 11 have difference 4.

Robert G. Wilson v, Table of n, a(n) for n = 0..2560 (first 501 terms from T. D. Noe)
Mark Herkommer, Goldbach Conjecture Research.
Tomás Oliveira e Silva, Goldbach conjecture verification.
The Prime Glossary, Goldbach's conjecture
+Plus Magazine ... living mathematics, Mathematical mysteries: the Goldbach conjecture
Eric Weisstein's World of Mathematics, Goldbach Conjecture
Wikipedia, Goldbach conjecture

Cf. A066285, A103147, records in A065978 and A066286.

f[n_] := For[p = n/2, True, p--, If[PrimeQ[p] && PrimeQ[n - p], Return[n/2 - p]]]; nn=101; t=Table[0,{nn}]; cnt=0; n=1; While[cnt

A066286 For even n>=4, let f(n)=A066285(n/2) be the minimal difference between primes p and q whose sum is n. This sequence contains the successive maxima of f.

Original entry on oeis.org

0, 2, 6, 18, 30, 36, 42, 54, 66, 78, 84, 90, 96, 150, 174, 186, 234, 240, 270, 276, 336, 366, 420, 456, 600, 666, 738, 786, 906, 1050, 1242, 1440, 1620, 1692, 2172, 2562, 2610, 3030, 3180, 3234, 3444, 3588, 3666, 3702, 4020, 4128, 4170, 4224, 4434, 4704, 5508, 5568, 6678, 6858, 8790, 8976, 10782

Offset: 0

Dean Hickerson, Dec 12 2001

nonn

All terms appear to be divisible by 6, except for the first two.

Robert G. Wilson v, Table of n, a(n) for n = 0..61 (terms 51..55 and 57 from Gilmar Rodriguez)

The corresponding values of n are in A065978.

f[n_] := For[p=n/2, True, p--, If[PrimeQ[p]&&PrimeQ[n-p], Return[n-2p]]]; For[n=4; max=-1, True, n+=2, If[f[n]>max, Print[max=f[n]]]]

a(51)-a(55) from Gilmar Rodriguez (Gilmar.Rodriguez(AT)nwfwmd.state.fl.us), Jun 16 2005

a(56) from Robert G. Wilson v, Jun 27 2005

A293858 Let n be even; m = n/2 and p a prime such that p<=m with n-p nonprime. The sequence contains the successive positive maxima of values n with L = primepi(m-1)-primepi(p+1)> 0.

Original entry on oeis.org

16, 44, 92, 148, 368, 400, 530, 688, 992, 1052, 2228, 3562, 4952, 7102, 10262, 20684, 37052, 52394, 61456, 62828, 80144, 224648, 236476, 251806, 360524, 362534, 742856, 1655152, 1872236, 2108282, 2319728, 2707118, 8561518, 12727966, 18115354, 18245438, 21572990, 54144704

Offset: 1

Gilmar Rodriguez Pierluissi, Oct 17 2017

nonn

Assuming the validity of Goldbach's Conjecture, there exists an integer L and a finite decreasing sequence of prime numbers P(i); i in {1,2,...,L}, such that P(L) < ... < P(2) < P(1) < m with n-P(i) not prime and n-P(L-1) prime, for P(L-1) prime.
The point {P(L-1), n-P(L-1)} is called the "minimal Goldbach point". The connotation of the word "minimal" is that this point lies on the line y = (-x + n) and sustains the shortest perpendicular distance to the line y = x, among all points {p,q} satisfying y=(-x+n) with prime p, 2 <= p <= m, such that n-p is prime.
Let L be the length of the set {P(1),P(2),..., P(L)}.
Notice that if m is prime then L=0.  Also; if n-P(1) is prime then L=0.

			For n=16, previous prime of m is 7; (n-7) is not prime; previous prime of 7 is 5; n-5 is prime; L=Length({7})=1.
For n=44, previous prime of m is 19; (n-19)is not prime; previous prime of 19 is 17; n-17 is not prime; previous prime of 17 is 13; (n-13) is prime; L=Length({19, 17})= 2.

Gilmar Rodriguez Pierluissi, Table of n, a(n) for n = 1..44
J.-M. Deshouillers, A. Granville, W. Narkiewicz and C. Pomerance, An upper bound in Goldbach's problem, Math. Comp. 61 (1993), 209-213.
Gilmar Rodriguez Pierluissi, Mathematica notebook (version 11.2) with examples for sequence A293858.
Gilmar Rodriguez Pierluissi, Adobe PDF file showing content of Mathematica notebook (in case that the reader does not have the Mathematica software available).

Cf. A065978.

PreviousPrime[n_]:=NextPrime[n, -1]
L[n_?EvenQ]:=Module[{m=n/2},If[PrimeQ[m],l=0,l=Length[Drop[Most@NestWhileList[PreviousPrime,m,!PrimeQ[n-#]&],1]]];l]
f[n_]:=For[m=n/2,True,m--,Return[L[n]]];For[n=16;max=-1,True,n+=2,If[f[n]>max,Print[n];max=f[n]]]

f(n) = {len = 0; m = n/2; if (isprime(m), return (0)); p = precprime(m-1); while (1, if (isprime(n-p), return (len)); p = precprime(p-1); len ++;);}
lista(nn) = {lmax = 0; forstep (n=2, nn, 2, newl = f(n); if (newl > lmax, print1(n, ", "); lmax = newl););} \\ Michel Marcus, Oct 22 2017

A307881 2a(n) is the least number where k sets a new record such that 2a(n)-k and 2a(n)+k are prime and at least one of 2a(n)-j and 2*a(n)+j is composite for all 0

Original entry on oeis.org

2, 4, 11, 23, 64, 68, 73, 119, 143, 172, 263, 452, 557, 868, 1238, 1579, 2864, 3533, 3637, 4252, 5171, 9263, 11282, 12388, 20036, 59119, 69332, 90131, 113783, 139283, 178612, 185714, 413788, 468059, 579932, 960707, 1879582, 2727031, 3266951, 3319868, 3591593

Offset: 1

Hugo Pfoertner, May 03 2019

nonn

The corresponding records of k are given in A307882.

Hugo Pfoertner, Table of n, a(n) for n = 1..53

Cf. A065978, A293858, A307882, A325142.

kmax=0; for(n=2,10^7,forstep(k=1,n,2,if(isprime(2*n-k)&&isprime(2*n+k),if(k>kmax,print1(n,", ");kmax=k);break(1))))

A307882 Records of the offset k in A307881, divided by 3.

Original entry on oeis.org

-1, 1, 3, 5, 7, 9, 11, 13, 15, 25, 29, 31, 39, 45, 61, 63, 75, 81, 85, 111, 123, 131, 151, 175, 207, 225, 241, 267, 301, 329, 335, 427, 435, 505, 539, 565, 611, 617, 665, 695, 739, 805, 843, 875, 1113, 1143, 1465, 1797, 1801, 1959, 2065, 2369, 2783

Offset: 1

Hugo Pfoertner, May 03 2019

The divisibility of k by 3 of all terms after a(1) is conjectural. Offsets not divisible by 3 are given as -k instead of k/3.

Cf. A065978, A293858, A307881, A325142.

kmax=0; for(n=2,10^7,forstep(k=1,n,2,if(isprime(2*n-k)&&isprime(2*n+k),if(k>kmax,if(k%3!=0,print1(-k,", "),print1(k/3,", "));kmax=k);break(1))))

cp's OEIS Frontend

A066285 a(n) is the minimal difference between primes p and q whose sum is 2n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A107926 The least number k such that there are primes p and q with p - q = 2*n, p + q = k, and p the least such prime >= k/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A066286 For even n>=4, let f(n)=A066285(n/2) be the minimal difference between primes p and q whose sum is n. This sequence contains the successive maxima of f.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A293858 Let n be even; m = n/2 and p a prime such that p<=m with n-p nonprime. The sequence contains the successive positive maxima of values n with L = primepi(m-1)-primepi(p+1)> 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A307881 2*a(n) is the least number where k sets a new record such that 2*a(n)-k and 2*a(n)+k are prime and at least one of 2*a(n)-j and 2*a(n)+j is composite for all 0

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A307882 Records of the offset k in A307881, divided by 3.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A307881 2a(n) is the least number where k sets a new record such that 2a(n)-k and 2a(n)+k are prime and at least one of 2a(n)-j and 2*a(n)+j is composite for all 0