A076304 -id:A076304 - cp's OEIS frontend

A226146 Numbers n such that n^2 is an average of three successive primes.

49, 161, 219, 351, 363, 469, 575, 597, 671, 877, 909, 933, 1013, 1225, 1231, 1303, 1359, 1381, 1419, 1489, 1577, 1653, 1797, 1815, 1989, 2083, 2117, 2177, 2241, 2289, 2301, 2403, 2483, 2493, 2517, 2611, 2617, 2653, 2727, 2779, 2869, 2931, 3029, 3051, 3261, 3515, 3617

Offset: 1

Alex Ratushnyak, May 28 2013

nonn

Cf. A076304, A206279, A226145-A226150.

#include 
#include 
#include 
#define TOP (1ULL<<30)
int main() {
  unsigned long long i, j, p1, p2, r, s;
  unsigned char *c = (unsigned char *)malloc(TOP/8);
  memset(c, 0, TOP/8);
  for (i=3; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0 /*&& i<(1ULL<<32)*/)
        for (j=i*i>>1; j>3] |= 1 << (j&7);
  for (p2=2, p1=3, i=5; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0) {
      s = p2 + p1 + i;
      if ((s%3)==0) {
        s/=3;
        r = sqrt(s);
        if (r*r==s) printf("%llu, ", r);
      }
      p2 = p1, p1 = i;
    }
  return 0;
}

Select[Sqrt[Mean[#]]&/@Partition[Prime[Range[10^6]],3,1],IntegerQ] (* Harvey P. Dale, Oct 23 2021 *)

A226147 Numbers n such that triangular(n) is an average of three successive primes.

Original entry on oeis.org

193, 233, 265, 301, 526, 709, 753, 922, 961, 962, 986, 1126, 1178, 1285, 1373, 1485, 1525, 1537, 1558, 1601, 1710, 1737, 1962, 1965, 2202, 2437, 2466, 2578, 2685, 2693, 2862, 3206, 3346, 3462, 3622, 3682, 3937, 3938, 3965, 4005, 4017, 4018, 4058, 4393, 4489, 4498, 4717

Offset: 1

Alex Ratushnyak, May 28 2013

nonn

Cf. A076304, A206279, A226145-A226150.

#include 
#include 
#include 
#define TOP (1ULL<<30)
int main() {
  unsigned long long i, j, p1, p2, r, s;
  unsigned char *c = (unsigned char *)malloc(TOP/8);
  memset(c, 0, TOP/8);
  for (i=3; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0 /*&& i<(1ULL<<32)*/)
        for (j=i*i>>1; j>3] |= 1 << (j&7);
  for (p2=2, p1=3, i=5; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0) {
      s = p2 + p1 + i;
      if ((s%3)==0) {
        s/=3;
        r = sqrt(s*2);
        if (r*(r+1)==s*2) printf("%llu, ", r);
      }
      p2 = p1, p1 = i;
    }
  return 0;
}

A123984 Primes p such that p^3 is a sum of three successive primes, or primes in A076306(n).

Original entry on oeis.org

11, 47, 223, 229, 313, 353, 397, 409, 571, 641, 661, 887, 1051, 1297, 1451, 1789, 2459, 2671, 2801, 2851, 3671, 4463, 4583, 4813, 4861, 5167, 5273, 5437, 5479, 5717, 5879, 6661, 6679, 6763, 6779, 7019, 7109, 7393, 7517, 7589, 7639, 7681, 7993, 8179, 8191, 9241

Offset: 1

Alexander Adamchuk, Oct 30 2006

nonn

A076306(n) = {11, 47, 145, 223, 229, 267, 313, 353, ...} Numbers n such that n^3 is a sum of three successive primes.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A076306, A076304. Cf. A122560 - Primes p such that p^2 is a sum of three successive primes. Cf. A122706 - Smallest prime p such that p^n is equal to the sum of 3 consecutive primes.

spQ[n_]:=Module[{n3=n^3,a,b,c,d,e},c=NextPrime[Floor[n3/3]];b=NextPrime[ c,-1];a=NextPrime[b,-1];d=NextPrime[c];e=NextPrime[d];n3==a+b+c || n3==b+c+d || n3==c+d+e];Select[Prime[Range[1200]],spQ] (* Harvey P. Dale, Sep 23 2011 *)

{ p1=prime(1) ; p2=prime(2) ; p3=prime(3) ; n3=p1+p2+p3 ; for(i=1,100000000, if( ispower(n3,3,&n), if(isprime(n), print(n) ) ; ) ; n3 -= p1 ; p1=p2 ; p2=p3 ; p3=nextprime(p3+1) ; n3 += p3 ; ) ; } \\ R. J. Mathar, Jan 13 2007

A000040 INTERSECT A076306. - R. J. Mathar, Jan 13 2007

More terms from R. J. Mathar, Jan 13 2007

a(15)-a(46) from Donovan Johnson, Apr 27 2008

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A226146 Numbers n such that n^2 is an average of three successive primes.

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A226147 Numbers n such that triangular(n) is an average of three successive primes.

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A123984 Primes p such that p^3 is a sum of three successive primes, or primes in A076306(n).

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