login
A054892
Smallest prime a(n) such that the sum of n consecutive primes starting with a(n) is divisible by n.
5
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
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 *)
KEYWORD
nonn
AUTHOR
Labos Elemer, May 23 2000
STATUS
approved