A122706 Smallest prime p such that p^n is equal to the sum of 3 consecutive primes.

23, 7, 11, 29, 79, 29, 509, 53, 467, 1571, 61, 7, 1553, 31, 1097, 11, 397, 11, 163, 677, 23, 103, 1723, 11, 1759, 67, 433, 149, 919, 2879, 293, 9907, 1103, 1153, 179, 6199, 2683, 1877, 4373, 4679, 953, 2341, 8069, 3779, 3691, 28463, 991, 1061, 2447, 5471

Offset

1,1

Comments

Corresponding numbers k such that a(n)^n = p(k) + p(k+1) + p(k+2) are given by A157197.

It is not known if a(n) exists for all n.

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..100

Formula

For m = (p^n)/3 (not an integer), if q,r are largest primes and s,t are smallest primes such that q < r < m < s < t, then p^n must equal either q+r+s or r+s+t. - Robert G. Wilson v

Example

a(1) = 23 because A070934(1) = p(3) + p(4) + p(5) = 5 + 7 + 11 = 23 is prime, but p(1) + p(2) + p(3) = 2 + 3 + 5 = 10 is composite and p(2) + p(3) + p(4) = 3 + 5 + 7 = 15 is composite.
a(2) = 7 because A122654(1) = 7 is prime and p(6) + p(7) + p(8) = 13 + 17 + 19 = 49 = 7^2, but p(k) + p(k+1) + p(k+2) are not squares for 0 < k < 6.

Mathematica

PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; f[n_] := Block[{p = If[n < 2, 5, 3]}, While[r = PrevPrim@ Floor[p^n/3]; q = PrevPrim@r; s = NextPrim@r; t = NextPrim@s; p^n != q + r + s && p^n != r + s + t, p = NextPrim@p]; p]; Array[f, 50] (* Robert G. Wilson v *)

Crossrefs

Cf. A070934, A122654.

Sequence in context: A040511 A264350 A058287 * A096640 A040510 A134910

Adjacent sequences: A122703 A122704 A122705 * A122707 A122708 A122709

Keyword

nonn

Author

Alexander Adamchuk, Sep 24 2006

Extensions

a(5)-a(50) from Robert G. Wilson v, Sep 26 2006

Status

approved