A090403 Balanced primes: Primes which are both the arithmetic mean and median of a sequence of 2k+1 consecutive primes, for some k>0.

5, 17, 29, 37, 53, 71, 79, 89, 137, 149, 151, 157, 173, 179, 193, 211, 227, 229, 257, 263, 281, 349, 353, 359, 373, 383, 397, 409, 419, 421, 433, 439, 487, 491, 563, 577, 593, 607, 631, 643, 653, 659, 677, 701, 709, 733, 751, 757, 787, 823, 827, 877, 947, 953

Offset

1,1

Comments

Union, for all k>0, of (2k+1)-balanced prime numbers, i.e., balanced prime of order k, which are primes p_n such that (2k+1)*p_n = Sum_{i=n-k..n+k} p_i, where p_i is the i-th prime.

Links

Donovan Johnson, Table of n, a(n) for n = 1..10000

Example

17 is in the sequence because 17 = (7 + 11 + 13 + 17 + 19 + 23 + 29)/7, (k = 3).
29 is in the sequence because 29 = (5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59)/15, (k = 7).
37 is a member because 37 = (7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71)/17; 7 & 71 are eight primes away from 37.

Mathematica

t[n_] := (For[k=1, !(SameQ[1/(2k+1)Sum[Prime[i], {i, n-k, n+k}], Prime[n]])&& k < n-1, k++ ]; k); b[n_] := If[t[n]<n-1||SameQ[1/(2n-1)Sum[Prime[i], {i, 2n-1}], Prime[n]], t[n], 0]; v={}; Do[If[b[n]!=0, v=Append[v, Prime[n]]], {n, 2, 168}]; v

Prog

(PARI) is_A090403(p)={my(s=0, n); isprime(p) & for(k=1, -1+n=primepi(p), (s+=prime(n+k)+prime(n-k)-2*p)||return(1); s>p & return)} \\ M. F. Hasler, Oct 21 2012

Crossrefs

Cf. A096693, A006562, A082077, A082078, A082079, A096697, A096698, A096699, A096700, A096701, A096702, A096703, A096704, A096711.

Sequence in context: A217512 A289836 A068829 * A096705 A023258 A030554

Adjacent sequences: A090400 A090401 A090402 * A090404 A090405 A090406

Keyword

easy,nonn

Author

Farideh Firoozbakht, Dec 07 2003

Extensions

Definition corrected by Franklin T. Adams-Watters, Apr 13 2006

Edited by M. F. Hasler, Oct 21 2012

Status

approved