This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(5) = 79 because the average of the 5 consecutive primes 71, 73, 79, 83, and 89 is 79, and this is the smallest such set: for example, the average of 7, 11, 13, 17, and 19 is 13.4, which is not an integer.
f[n_] := Block[{k = 1, t},While[t = Table[Prime[i], {i, k, k + n - 1}]; Mod[Plus @@ t, n] > 0, k++ ];t];Mean /@ Table[f[n], {n, 58}] (* Ray Chandler, Oct 09 2006 *)
a(n) = {my(v=primes(n), s=vecsum(v), p=prime(n)); while(s%n, s-=v[1]-p=nextprime(p+1); v=concat(v[2..n], p)); s/n; } \\ Jinyuan Wang, Sep 05 2020
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.
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 *)
Triangle begins: 2 3 5 3 5 7 5 7 11 13 71 73 79 83 89 5 7 11 13 17 19 7 11 13 17 19 23 29 17 19 23 29 31 37 41 43 239 241 251 257 263 269 271 277 281 13 17 19 23 29 31 37 41 43 47 29 31 37 41 43 47 53 59 61 67 71 5 7 11 13 17 19 23 29 31 37 41 43
f[n_] := Block[{k = 1, t},While[t = Table[Prime[i], {i, k, k + n - 1}]; Mod[Plus @@ t, n] > 0, k++ ];t];Flatten[Table[f[n], {n, 12}]]
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