A231576 -id:A231576 - 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 *)

A231578 Sequence of pairs k,g such that k is the smallest and k2^n+1-g, k2^n+1, k*2^n+1+g are three consecutive primes in arithmetic progression.

Original entry on oeis.org

105, 12, 1, 2, 219, 6, 37, 6, 245, 12, 109, 6, 39, 6, 1, 6, 245, 12, 669, 6, 9, 6, 45, 12, 5, 12, 1413, 6, 297, 6, 37, 6, 2019, 18, 603, 24, 467, 6, 897, 6, 303, 6, 1687, 12, 77, 30, 1437, 6, 711, 36, 2293, 6, 315, 42, 525, 12, 1799, 12, 4137, 30, 1019, 6

Offset: 1

Pierre CAMI, Nov 11 2013

nonn

			1*2^8+1-6=251, 1*2^8+1=257, 1*2^8+1+6=263, so the pair for n=8 is 1,6.

Pierre CAMI, Table of n, a(n) for n = 1..1300
Pierre CAMI, PFGW Script

Cf. A231576.

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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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A231578 Sequence of pairs k,g such that k is the smallest and k2^n+1-g, k2^n+1, k*2^n+1+g are three consecutive primes in arithmetic progression.

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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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A231578 Sequence of pairs k,g such that k is the smallest and k*2^n+1-g, k*2^n+1, k*2^n+1+g are three consecutive primes in arithmetic progression.

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A231578 Sequence of pairs k,g such that k is the smallest and k2^n+1-g, k2^n+1, k*2^n+1+g are three consecutive primes in arithmetic progression.