A054892 Smallest prime a(n) such that the sum of n consecutive primes starting with a(n) is divisible by n.

2, 3, 3, 5, 71, 5, 7, 17, 239, 13, 29, 5, 43, 23, 5, 5, 7, 7, 79, 17, 47, 11, 2, 73, 97, 53, 271, 13, 263, 23, 41, 61, 97, 101, 181, 41, 47, 13, 233, 13, 53, 13, 359, 151, 71, 61, 239, 73, 443, 859, 29, 131, 2, 61, 313, 101, 19, 151, 521, 3, 571, 31, 7, 79, 109, 97, 53

Offset

1,1

Comments

See A132809 for another version.

In some cases (n=1,2,25,..), like a(25)=97, the sum of 25 consecutive primes starts with the 25th prime and is divided by 25: Sum=97+...+227=3925=25*157

Links

Zak Seidov, Table of n, a(n) for n = 1..1000

Formula

Min[q_1; Sum[q_i; {i, 1, n}]]=n*X], q_i is a prime (rarely only q_i=Prime[i])

Example

a(8) = 17 since the sum of the 8 consecutive primes starting with 17 is 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 = 240, which is divisible by 8.  No prime less than 17 has this property: for example, 7 + 11 + ... + 31 = 150 which is not divisible by 8.

Mathematica

f[n_] := Block[{k = 1, t}, While[t = Table[Prime[i], {i, k, k + n - 1}]; Mod[Plus @@ t, n] > 0, k++ ]; t]; First /@ Table[f[n], {n, 67}] (* Ray Chandler, Oct 09 2006 *)
Module[{prs=Prime[Range[250]]}, Table[SelectFirst[Partition[prs, n, 1], Mod[Total[#], n]==0&], {n, 70}]][[;; , 1]] (* Harvey P. Dale, Jul 11 2023 *)

Crossrefs

Cf. A054643, A024672, A034961, A077388, A077389, A122820, A132809.

Sequence in context: A355868 A046826 A323713 * A104570 A103356 A345385

Adjacent sequences: A054889 A054890 A054891 * A054893 A054894 A054895

Keyword

nonn

Author

Labos Elemer, May 23 2000

Status

approved