Gilmar Rodriguez Pierluissi - cp's OEIS frontend

User: Gilmar Rodriguez Pierluissi

Gilmar Rodriguez Pierluissi's wiki page.

Gilmar Rodriguez Pierluissi has authored 4 sequences.

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.

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

A109885 Let n be an even integer > 2. Let PrimeP be the number of prime partition pairs {p,q} corresponding to n such that n = p + q, p and q are prime and p <= q. Let CompP be the number of composite partition pairs {r,s} corresponding to n such that n = r + s, r is prime, s is composite and r <= s. For what n's is 2*PrimeP > CompP?

Original entry on oeis.org

4, 10, 22, 24, 34, 36, 48, 54, 60, 66, 72, 78, 84, 90, 102, 114, 120, 126, 144, 150, 156, 168, 180, 186, 198, 204, 210, 240, 246, 252, 270, 294, 300, 324, 330, 360, 378, 390, 420, 450, 462, 480, 510, 540, 546, 570, 600, 630, 660, 690, 714, 720, 750, 780, 840

Offset: 1

Gilmar Rodriguez Pierluissi (gilmarlily(AT)yahoo.com), Aug 31 2005

nonn

Except for a(1), a(2) a(3) & a(5), a(n)==0 (mod 6). - Robert G. Wilson v

fQ[n_] := Block[{t = n - Prime@Range@PrimePi[n/2]}, 2Length[Select[t, PrimeQ]] > Length[t]]; Select[ 2Range[2, 434], fQ[ # ] &] (* Robert G. Wilson v, Nov 03 2005 *)

Edited by Robert G. Wilson v, Nov 03 2005

A109440 Even numbers k such that exactly half of the primes p with p <= k/2 give k-p also a prime.

Original entry on oeis.org

2, 6, 8, 14, 16, 18, 20, 26, 30, 42, 108, 132, 234, 264, 288, 354, 504, 1920, 2010, 5040

Offset: 1

Gilmar Rodriguez Pierluissi (gilmarlily(AT)yahoo.com), Aug 26 2005

Let k be an even integer. Let PrimeP be the set of prime partition points {p,q} such that p + q = k, p and q are prime and p <= q. Let CompP be the set of composite partition points {p,q} such that p + q = k, p is prime, q is composite and p <= q. Sequence gives value of k such that the size of the two sets PrimeP and CompP are equal.

Even numbers k such that 2*A045917(k/2) = A000720(k/2). - Andrew Howroyd, Jan 02 2020

			The number k = 30 is included because the two sets PrimeP={{7,23},{11,19},{13,17}} and CompP={{2,28},{3,27},{5,25}} have the same number of elements.

Cf. A000720, A045917, A109885.

d(n)={my(s=0); forprime(p=2, n, s+=if(isprime(2*n-p), 1, -1)); s}
{ for(n=1, 10^4/2, if(d(n)==0, print1(2*n, ", "))) } \\ Andrew Howroyd, Jan 02 2020

Name edited and a(13)-a(20) from Andrew Howroyd, Jan 02 2020

A109679 Smallest even number which is the unordered sum of two primes in more ways than any previous even number.

Original entry on oeis.org

2, 4, 10, 22, 34, 48, 60, 78, 84, 90, 114, 120, 168, 180, 210, 300, 330, 390, 420, 510, 630, 780, 840, 990, 1050, 1140, 1260, 1470, 1650, 1680, 1890, 2100, 2310, 2730, 3150, 3570, 3990, 4200, 4410, 4620, 5250, 5460, 6090, 6510, 6930, 7980, 8190, 9030, 9240

Offset: 1

Gilmar Rodriguez Pierluissi (gilmarlily(AT)yahoo.com), Aug 30 2005

nonn

Record value of A023036 or A045917.

a(n)== 0 (mod 30) for n > 13.

Vincenzo Librandi, Table of n, a(n) for n = 1..150
Eric Weisstein's World of Mathematics, Goldbach Partition
Index entries for sequences related to Goldbach conjecture

Essentially the same as A082917. Cf. A082918, A002375, A023036, A045917, A000954.

f[n_] := Length[ Select[n - Prime@ Range@ PrimePi[n/2], PrimeQ]]; t = {}; mxm = -1; Do[ If[ f[n] > mxm, AppendTo[t, n]; mxm = f[n]], {n, 2, 9000, 2}]; t

Edited and extended by Robert G. Wilson v, Sep 08 2005

Changed offset from 0 to 1 by Vincenzo Librandi, Apr 18 2013

cp's OEIS Frontend

User: Gilmar Rodriguez Pierluissi

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

Views

Author

Keywords

Comments

Programs

Mathematica

Extensions

A109440 Even numbers k such that exactly half of the primes p with p <= k/2 give k-p also a prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A109679 Smallest even number which is the unordered sum of two primes in more ways than any previous even number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions