A035789
Start of a string of exactly 1 consecutive (but disjoint) pair of twin primes.
Original entry on oeis.org
29, 41, 59, 71, 227, 239, 269, 281, 311, 347, 461, 521, 569, 599, 617, 641, 659, 857, 881, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1607, 1619, 1667, 1697, 1721, 1787, 1997, 2027, 2141, 2237, 2267, 2309, 2339, 2381, 2549, 2591, 2657, 2687
Offset: 1
The first lonely twin primes (A069453) are 29,31 (23 and 37 are non-twin), 41,43 (37 and 47 are non-twin), 59,61 (53 and 67 are non-twin). Of these, the lesser twins are 29,41,59, so this is how the sequence begins.
23, 27, 29, 31, 37, 41: 27-23>2, 31-29=2, 41-37>2; so 29 is in the sequence.
From _Hartmut F. W. Hoft_, Apr 05 2016: (Start)
The example should read: 19, 23, 29, 31, 37, 41: 23-19>2, 31-29=2, 41-37>2; so 29 is in the sequence.
a(n)=A069453(2n-1), n>=1.
(End)
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PrimeNext[n_]:=Module[{k},k=n+1;While[ !PrimeQ[k],k++ ];k]; PrimePrev[n_]:=Module[{k},k=n-1;While[ !PrimeQ[k],k-- ];k]; lst={};Do[p=Prime[n];If[ !PrimeQ[p-2]&&!PrimeQ[p+4]&&PrimeQ[p+2]&&!PrimeQ[PrimePrev[p]-2]&&!PrimeQ[PrimeNext[p+2]+2],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 22 2009 *)
(* starting at n=3 would eliminate the first two primality tests, Hartmut F. W. Hoft, Apr 09 2016 *)
A069455
Greater of lonely twin primes.
Original entry on oeis.org
31, 43, 61, 73, 229, 241, 271, 283, 313, 349, 463, 523, 571, 601, 619, 643, 661, 859, 883, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1609, 1621, 1669, 1699, 1723, 1789, 1999, 2029, 2143, 2239, 2269, 2311, 2341, 2383
Offset: 1
The first lonely twin primes (A069453) are 29,31 (23 and 37 are non-twin), 41,43 (37 and 47 are non-twin), 59,61 (53 and 67 are non-twin). Of these, the greater twins are 31,43,61, so this is how the sequence begins.
A069456
Non-twin primes that are at least doubly lonely.
Original entry on oeis.org
1039, 2099, 4253, 91121, 386401, 626617, 754973, 873553, 908857, 972137, 1619353, 1749067, 1841681, 2007899, 2169007, 2241353, 2420633, 2484931, 2594971, 3075323, 3129601, 3151843, 3837451, 3843247, 3919229, 4038709, 4545683, 5502449, 5530529, 5921869
Offset: 1
These are non-twin primes sandwiched between at least 2 pairs of twins on each side. The first number in the sequence is 1039 (sandwiched between 1019,1021,1031,1033 and 1049,1051,1061,1063).
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Primes:= select(isprime,[seq(i,i=3..6*10^6,2)]):
good:= select(t -> Primes[t-3]-Primes[t-4]=2 and Primes[t-1]-Primes[t-2]=2 and Primes[t+2]-Primes[t+1]=2 and Primes[t+4]-Primes[t+3]=2, [$5..nops(Primes)-4]):
Primes[good]; # Robert Israel, May 13 2016
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dltpQ[{a_,b_,c_,d_,e_,f_,g_,h_,i_}]:=b-a==d-c==g-f==i-h==2; Transpose[ Select[ Partition[Prime[Range[410000]],9,1],dltpQ]][[5]] (* Harvey P. Dale, May 14 2013 *)
A262935
Increasing distances of lonely twin primes pairs to nearest prime.
Original entry on oeis.org
1, 2, 4, 6, 10, 12, 16, 18, 28, 30, 34, 42, 46, 48, 58, 88, 90, 94, 124, 130, 136, 154, 162, 168, 172, 178, 202, 216, 258, 264, 294, 342, 352, 354, 364, 366, 370, 378, 396, 408
Offset: 1
(3,5) is a twin primes pair, min(7-5, 3-2)=1, therefore a(1)=1.
(5,7) is a twin primes pair, min(11-7, 5-3)=2>1, therefore a(2)=2.
(11,13) is a twin primes pair, min(17-13, 11-7)=4>2, therefore a(3)=4.
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{m=0; q=5; s=3; t=2; forprime(p=6, 10^9, if((q-s==2) && (min(p-q, s-t)>m), m=min(p-q, s-t); print1(m, ", ") ); t=s; s=q; q=p;)}
A068016
Lonely non-twin primes: non-twins sandwiched between two pairs of twins.
Original entry on oeis.org
23, 37, 67, 233, 277, 631, 1039, 1283, 1297, 1307, 1613, 1693, 1709, 2099, 2137, 2333, 2719, 2797, 3271, 3533, 3547, 3571, 3923, 4027, 4253, 4523, 4643, 4793, 5483, 5507, 5647, 6563, 7321, 8831, 8849, 9007, 9029, 10061, 10079, 10289, 10513, 12049, 13687
Offset: 1
37 is in the sequence because it is adjacent to two pairs of twins (29,31 and 41,43). 47 is not because the primes adjacent to it are 43 and 53 and although 43 is a twin, 53 is not.
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PrimeNext[n_]:=Module[{k},k=n+1;While[ !PrimeQ[k],k++ ];k]; PrimePrev[n_]:=Module[{k},k=n-1;While[ !PrimeQ[k],k-- ];k]; lst={};Do[p=Prime[n];If[ !PrimeQ[p-2]&&PrimeQ[PrimePrev[p]-2]&&!PrimeQ[p+2]&&PrimeQ[PrimeNext[p]+2],AppendTo[lst,p]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 22 2009 *)
m = 2000; #[[3]] & /@ Select[Partition[Prime[Range[7, m]], 5, 1], #[[2]] - #[[1]] == #[[5]] - #[[4]] == 2 &] (* Zak Seidov, Nov 25 2012 *)
A069479
Smallest n-tuply-lonely non-twin prime.
Original entry on oeis.org
23, 1039, 403026797, 121829611399
Offset: 1
An n-tuply lonely non-twin prime is a non-twin prime that is sandwiched between exactly n pairs of prime twins on both sides. a(2)=1039 because 1039 is the first non-twin to be sandwiched between exactly 2 pairs of twins on each side (1019,1021,1031,1033 and 1049,1051,1061,1063).
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