A233742 -id:A233742 - cp's OEIS frontend

A233546 Smallest k such that 6^n+k, 6^n+k+g, 6^n+k+2*g are consecutive primes in arithmetic progression.

11, 35, 65, 41, 221, 655, 515, 263, 4265, 893, 4085, 1031, 3161, 145, 821, 2083, 2101, 433, 3743, 2243, 511, 2623, 5653, 271, 2885, 4157, 18023, 9023, 1151, 4787, 737, 2141, 2833, 6181, 3217, 3635, 715, 4501, 5381, 4231, 13265, 823

Offset: 2

Pierre CAMI, Dec 12 2013

nonn

The sequence starts at n=2 as there is no solution for n=1.

The primes are probable primes for n>23.

			6^2+11=47, 6^2+11+6=53, 6^2+11+2*6=59 and 47, 53, 59 are consecutive primes
and k=11 is minimal (since although 6^2+5=41, 6^2+5+6=47, 6^2+5+2*6=53 are primes, they are not consecutive primes), so a(2)=11. (Example clarified by _Jonathan Sondow_, Dec 16 2013.)

Pierre CAMI, Table of n, a(n) for n = 2..350
Wikipedia, Consecutive primes in arithmetic progression

Cf. A231576, A231578, A233550, A233742.

a[n_] := For[k = 1, True, k = k+2, p = 6^n+k; If[PrimeQ[p], q = NextPrime[p]; r = NextPrime[q]; g = q-p; If[g == r-q, Print["n = ", n, " k = ", k, " g = ", g, " ", {p, q, r}]; Return[k]]]]; Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Dec 17 2013 *)

A233550 Gap g between 3 consecutive primes for the smallest k such that 6^n+k, 6^n+k+g, 6^n+k+2*g are consecutive primes in arithmetic progression.

Original entry on oeis.org

6, 6, 6, 6, 12, 6, 18, 12, 6, 24, 18, 12, 6, 18, 12, 42, 30, 12, 54, 24, 60, 30, 24, 36, 78, 18, 42, 132, 42, 24, 24, 60, 24, 72, 24, 36, 30, 6, 12, 30, 30, 120, 6, 36, 72, 30, 30, 18, 6, 60, 210, 66, 84, 30, 96, 24, 84, 6, 210, 78, 18, 228

Offset: 2

Pierre CAMI, Dec 16 2013

nonn

Sequence starts for n=2 as no solution for n=1.

g is a multiple of 6 as otherwise 6^n+k, 6^n+k+g, or 6^n+k+2*g is divisible by 2 or 3. - Jonathan Sondow, Dec 16 2013

			6^2+11=47, 6^2+11+6=53, 6^2+11+2*6=59 are consecutive primes and k=11 is minimal, so a(2)=6. - _Jonathan Sondow_, Dec 16 2013

Pierre CAMI, Table of n, a(n) for n = 2..350

Cf. A233546 (associated k), A233742.

PFGW
```
; See A233546.
```

a(n) = 6*A233742(n). - Jonathan Sondow, Dec 16 2013

A233823 Least prime in arithmetic progression of consecutive primes 6^n+k, 6^n+k+g, 6^n+k+2*g with smallest k.

Original entry on oeis.org

47, 251, 1361, 7817, 46877, 280591, 1680131, 10077959, 60470441, 362797949, 2176786421, 13060695047, 78364167257, 470184984721, 2821109908277, 16926659446819, 101559956670517, 609359740010929, 3656158440066719, 21936950640380099, 131621703842267647, 789730223053605439

Offset: 2

Jonathan Sondow, Dec 16 2013

nonn

No solution for n = 1, so sequence starts at n = 2.

The primes are probable primes for n > 23.

			a(2) = 6^2 + A233546(2) = 36 + 11 = 47.

Cf. A233546 (values of k), A233550 (values of g), A233742 (= g/6).

a(n) = 6^n + A233546(n).

cp's OEIS Frontend

A233546 Smallest k such that 6^n+k, 6^n+k+g, 6^n+k+2*g are consecutive primes in arithmetic progression.

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A233550 Gap g between 3 consecutive primes for the smallest k such that 6^n+k, 6^n+k+g, 6^n+k+2*g are consecutive primes in arithmetic progression.

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A233823 Least prime in arithmetic progression of consecutive primes 6^n+k, 6^n+k+g, 6^n+k+2*g with smallest k.

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