Any prime number is the average of at least one pairs of inequal prime numbers

According to Goldbach’s conjecture, any 2*p = p + p fulfills it. This can be a little stronger, to make 2*p=p1+p2, p1 <> p2, it still stands.

For example:

“Prime[5]:11+/-6*1=5+17”
“Prime[6]:13+/-6*1=7+19”
“Prime[7]:17+/-6*1=11+23”
“Prime[8]:19+/-6*2=7+31”
“Prime[9]:23+/-6*1=17+29”
“Prime[10]:29+/-6*2=17+41”
“Prime[11]:31+/-6*2=19+43”
“Prime[12]:37+/-6*1=31+43”
“Prime[13]:41+/-6*2=29+53”
“Prime[14]:43+/-6*4=19+67”

The full list up to:

“Prime[1306583450]:30158696947+/-6*185=30158695837+30158698057”

is downloadable as GoldbachAverage_typed_ranks.primes.tgz (caution, large file).

It is straight forward that only when k=6i, it is possible both p+k and p-k are both prime.

Assuming for prime p, both p+6i and p-6i are prime, the first occurance of i forms the integer sequence A139602, named:

“a(n) is the smallest prime p that makes the pair p+/-6n both primes while no other pair of p+/-6k, 0<k<n primes “.

Defining a(n) = A139602(m) such that for any k>m A139602(k) > A139602(m), the first 65 items are listed below:
“Prime[5]:11+/-6*1=5+17”
“Prime[8]:19+/-6*2=7+31”
“Prime[14]:43+/-6*4=19+67”
“Prime[25]:97+/-6*5=67+127”
“Prime[38]:163+/-6*6=127+199”
“Prime[43]:191+/-6*7=149+233”
“Prime[48]:223+/-6*10=163+283”
“Prime[88]:457+/-6*14=373+541”
“Prime[151]:877+/-6*15=787+967”
“Prime[176]:1049+/-6*17=947+1151”
“Prime[214]:1307+/-6*20=1187+1427”
“Prime[300]:1987+/-6*21=1861+2113”
“Prime[308]:2029+/-6*25=1879+2179”
“Prime[320]:2129+/-6*30=1949+2309”
“Prime[577]:4217+/-6*34=4013+4421”
“Prime[853]:6599+/-6*45=6329+6869”
“Prime[1228]:9967+/-6*51=9661+10273”
“Prime[1271]:10357+/-6*79=9883+10831”
“Prime[2090]:18233+/-6*81=17747+18719”
“Prime[6615]:66343+/-6*89=65809+66877”
“Prime[7356]:74573+/-6*111=73907+75239”
“Prime[9243]:95911+/-6*113=95233+96589”
“Prime[9568]:99719+/-6*132=98927+100511”
“Prime[16880]:186551+/-6*133=185753+187349”
“Prime[17686]:196337+/-6*135=195527+197147”
“Prime[18911]:211219+/-6*157=210277+212161”
“Prime[23026]:262469+/-6*160=261509+263429”
“Prime[24229]:277301+/-6*163=276323+278279”
“Prime[35125]:416573+/-6*175=415523+417623”
“Prime[49360]:603487+/-6*211=602221+604753”
“Prime[78101]:994549+/-6*222=993217+995881”
“Prime[107328]:1403137+/-6*271=1401511+1404763”
“Prime[290914]:4117441+/-6*273=4115803+4119079”
“Prime[335833]:4805761+/-6*290=4804021+4807501”
“Prime[341710]:4895789+/-6*307=4893947+4897631”
“Prime[401477]:5823067+/-6*309=5821213+5824921”
“Prime[402723]:5842813+/-6*341=5840767+5844859”
“Prime[521180]:7704409+/-6*385=7702099+7706719”
“Prime[965375]:14911571+/-6*390=14909231+14913911”
“Prime[1041561]:16174121+/-6*427=16171559+16176683”
“Prime[1403631]:22245077+/-6*460=22242317+22247837”
“Prime[2706070]:44786009+/-6*472=44783177+44788841”
“Prime[3165153]:52912507+/-6*569=52909093+52915921”
“Prime[9066474]:161738579+/-6*627=161734817+161742341”
“Prime[11872208]:215189881+/-6*632=215186089+215193673”
“Prime[13761571]:251589509+/-6*772=251584877+251594141”
“Prime[37548968]:726419297+/-6*791=726414551+726424043”
“Prime[45509717]:889697437+/-6*805=889692607+889702267”
“Prime[70323838]:1407132329+/-6*833=1407127331+1407137327”
“Prime[73701270]:1478355583+/-6*855=1478350453+1478360713”
“Prime[91179428]:1849422983+/-6*864=1849417799+1849428167”
“Prime[105481347]:2155728167+/-6*899=2155722773+2155733561”
“Prime[114690246]:2354047967+/-6*980=2354042087+2354053847”
“Prime[126391595]:2607147923+/-6*986=2607142007+2607153839”
“Prime[149035786]:3100157353+/-6*994=3100151389+3100163317”
“Prime[179785305]:3775276711+/-6*1000=3775270711+3775282711”
“Prime[197058908]:4157064299+/-6*1098=4157057711+4157070887”
“Prime[227165893]:4826177711+/-6*1145=4826170841+4826184581”
“Prime[416967557]:9124822669+/-6*1150=9124815769+9124829569”
“Prime[426241552]:9337604803+/-6*1166=9337597807+9337611799”
“Prime[460241741]:10119526379+/-6*1213=10119519101+10119533657”
“Prime[544180839]:12060924103+/-6*1219=12060916789+12060931417”
“Prime[699679610]:15691824767+/-6*1279=15691817093+15691832441”
“Prime[1083294128]:24791915689+/-6*1292=24791907937+24791923441”
“Prime[1295780294]:29898040813+/-6*1385=29898032503+29898049123”