A219558 Number of odd prime pairs {p,q} (p>q) such that p+(1+(n mod 2))q=n and ((p-1-(n mod 2))/q)=((q+1)/p)=1 where (-) denotes the Legendre symbol.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 3, 0, 2, 0, 1, 1, 1, 1, 2, 2, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 3, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 2, 0, 0, 2, 1, 2, 1, 1, 0, 1, 1, 2, 2, 3, 0, 0, 0, 0

Offset

1,24

Comments

For any integer m, define s(m) as the smallest positive integer s such that for each n=s,s+1,... there are primes p>q>2 with p+(1+(n mod 2))q=n and ((p-(1+(n mod 2))m)/q)=((q+m)/p)=1. If such a positive integer s does not exist, then we set s(m)=0.

Zhi-Wei Sun has the following general conjecture: s(m) is always positive. In particular, s(0)=1239,

s(1)=1470, s(-1)=2192, s(2)=1034, s(-2)=1292,

s(3)=1698, s(-3)=1788, s(4)=848, s(-4)=1458,

s(5)=1490, s(-5)=2558, s(6)=1115, s(-6)=1572,

s(7)=1550, s(-7)=932, s(8)=825, s(-8)=2132,

s(9)=1154, s(-9)=1968, s(10)=1880, s(-10)=1305,

s(11)=1052, s(-11)=1230, s(12)=2340, s(-12)=1428,

s(13)=2492, s(-13)=2673, s(14)=1412, s(-14)=1638,

s(15)=1185, s(-15)=1230, s(16)=978, s(-16)=1605,

s(17)=1154, s(-17)=1692, s(18)=1757, s(-18)=2292,

s(19)=1230, s(-19)=2187, s(20)=2048, s(-20)=1372,

s(21)=1934, s(-21)=1890, s(22)=1440, s(-22)=1034,

s(23)=1964, s(-23)=1322, s(24)=1428, s(-24)=2042,

s(25)=1734, s(-25)=1214, s(26)=1260, s(-26)=1230,

s(27)=1680, s(-27)=1154, s(28)=1652, s(-28)=1808,

s(29)=1112, s(-29)=1670, s(30)=1820, s(-30)=1284.

Note that s(1)=1470 means that a(n)>0 for all n=1470,1471,... That s(0)=1239 is related to a conjecture of Olivier Gérard and Zhi-Wei Sun.

If we replace ((p-1-(n mod 2))/q)=((q+1)/p)=1 in the definition of a(n) by ((p-1)/q)=((q+1)/p)=1, then the new a(n) seems positive for any n>1181.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Olivier Gérard and Zhi-Wei Sun, Refining Goldbach's conjecture by using quadratic residues, a message to Number Theory List, Nov. 19, 2012.

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Example

a(14)=1 since 14=11+3 with ((11-1)/3)=((3+1)/11)=1.
a(31)=1 since 31=17+2*7 with ((17-2)/7)=((7+1)/17)=1.

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[n-(1+Mod[n, 2])Prime[k]]==True&&JacobiSymbol[n-(1+Mod[n, 2])(Prime[k]+1), Prime[k]]==1&&JacobiSymbol[Prime[k]+1, n-(1+Mod[n, 2])Prime[k]]==1, 1, 0], {k, 2, PrimePi[(n-1)/(2+Mod[n, 2])]}]
Do[Print[n, " ", a[n]], {n, 1, 10000}]

Crossrefs

Cf. A002375, A046927, A219055, A219157, A218867, A219185, A218754, A218825, A219052.

Sequence in context: A285640 A231189 A363857 * A279210 A232243 A034876

Adjacent sequences: A219555 A219556 A219557 * A219559 A219560 A219561

Keyword

nonn

Author

Zhi-Wei Sun, Nov 23 2012

Status

approved