A051795 -id:A051795 - cp's OEIS frontend

A096710 Quadruply balanced primes: primes which are averages of their immediate neighbor primes, their second neighbor primes, their third neighbor primes and their fourth neighbor primes.

98303927, 580868459, 784857323, 857636141, 909894647, 951508837, 1367470823, 1480028171, 1850590099, 2106973159, 2121382079, 2409718043, 2635873907, 2704854637, 3225527099, 3386231579, 3823510039, 3824915671, 3905211517, 4123167667, 4127991383, 4386448117

Offset: 1

Robert G. Wilson v, Jun 28 2004

nonn

			98303927 is a member because 98303927 = (98303903 + 98303951)/2 = (98303897 + 98303957)/2 = (98303873 + 98303981)/2 = (98303867 + 98303987)/2.

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

Cf. A006562, A051795, A096709, A055380.

a = {}; Do[p = 2Prime[n]; If[p == Prime[n - 1] + Prime[n + 1], If[ p == Prime[n - 2] + Prime[n + 2], If[p == Prime[n - 3] + Prime[n + 3], If[p == Prime[n - 4] + Prime[n + 4], Print[p/2]; AppendTo[a, p/2]]]]], {n, 6, 117039731}]; a
Select[Partition[Prime[Range[207405000]],9,1],(#[[1]]+#[[9]])/2 == (#[[2]]+ #[[8]])/2 == (#[[3]]+#[[7]])/2==(#[[4]]+#[[6]])/2==#[[5]]&][[All,5]] (* Harvey P. Dale, Dec 27 2018 *)

More terms from Jud McCranie, Sep 29 2006

A126554 Arithmetic mean of two consecutive balanced primes (of order one).

Original entry on oeis.org

29, 105, 165, 192, 234, 260, 318, 468, 578, 600, 630, 693, 840, 962, 1040, 1113, 1155, 1205, 1295, 1439, 1629, 1750, 1830, 2097, 2352, 2547, 2790, 2933, 3135, 3310, 3475, 3685, 3873, 4211, 4433, 4527, 4627, 4674, 4842, 5050, 5110, 5208, 5345, 5390, 5478

Offset: 1

Artur Jasinski, Dec 27 2006

nonn

Might be called interprimes of order two, since the arithmetic means of two consecutive odd primes (A024675) sometimes are called interprimes.
Balanced primes of order two (A082077) and doubly balanced primes (A051795) have different definitions.
For primes in this sequence (prime interprimes of order two) see A126555.

Muniru A Asiru and Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Muniru A Asiru)

Cf. A006562, A024675, A082077, A051795, A126555.

P:=Filtered([1..6000],IsPrime);;P1:=List(Filtered(List([0..Length(P)-3],k->List([1..3],j->P[j+k])),i->Sum(i)/3=i[2]),m->m[2]);;
a:=List([1..Length(P1)-1],n->(P1[n+1]+P1[n])/2); # Muniru A Asiru, Mar 31 2018

b = {}; a = {}; Do[If[PrimeQ[((Prime[n + 2] + Prime[n + 1])/2 + (Prime[n + 1] + Prime[n])/2)/2], AppendTo[a, ((Prime[n + 2] + Prime[n + 1])/2 + (Prime[n + 1] + Prime[n])/2)/2]], {n, 1, 1000}]; Do[AppendTo[b, (a[[k + 1]] + a[[k]])/2], {k, 1, Length[a] - 1}]; b

{m=6000;a=0;p=2;q=3;r=5;while(r<=m,if((p+r)/2==q,if(a>0,print1((a+q)/2,","));a=q);p=q;q=r;r=nextprime(r+1))} \\ Klaus Brockhaus, Jan 05 2007

a(n) = (A006562(n+1)+A006562(n))/2.

Edited by Klaus Brockhaus, Jan 05 2007

A082312 Half the difference between start and center prime of the smallest [2n+1]-balanced prime set (A082080).

Original entry on oeis.org

1, 4, 5, 14, 11, 14, 12, 15, 32, 36, 32, 30, 41, 65, 42, 41, 53, 45, 75, 76, 69, 63, 99, 98, 60, 112, 99, 84, 94, 130, 132, 103, 87, 140, 172, 175, 144, 190, 171, 140, 200, 145, 203, 190, 155, 168, 202, 210, 144, 157, 254, 185, 189, 306, 201, 323, 303, 229, 267

Offset: 1

Ralf Stephan, Apr 09 2003

nonn

			The smallest 5-balanced prime, 79 (center of 71,73,79,83,89) minus 8 is 71, so a(2)=8/2=4.

Cf. A006562, A075540, A054800, A051795.

for(n=1, 80, i=2*n+1; f=0; forprime(p=2, 10^7, s=0; c=i; pr=p-1; t=0; while(c>0, c=c-1; pr=nextprime(pr+1); s=s+pr; if(c==(i-1)/2, t=pr)); if(s/i==t, print1((t-p)/2", "); f=1; break)); if(!f, print1("0, ")))

A160920 Primes which are at the same time balanced primes of order 2, 3 and 4.

Original entry on oeis.org

236429, 1108477, 1829801, 2073263, 2191513, 2192789, 3236267, 3990031, 4248947, 4485683, 4986061, 6869969, 7711079, 8473811, 8480911, 9282173, 9327277, 9350123, 9547303, 9730649, 12077909, 12993917, 13165441, 13398611, 14129761, 14785907

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 30 2009

nonn

The intersection of A082077, A082078 and A082079.

Muniru A Asiru, Table of n, a(n) for n = 1..200
Wikipedia, Balanced prime

Cf. A040040, A051795, A082077, A082078, A082079.

P:=Filtered([1,3..2*10^7+1],IsPrime);;
a:=Intersection(List([2,3,4],b->List(Filtered(List([0..Length(P)-(2*b+1)],k->List([1..2*b+1],j->P[j+k])),i->Sum(i)/(2*b+1)=i[b+1]),m->m[b+1]))); # Muniru A Asiru, Apr 08 2018

isBalPr := proc(p,o) local r,s,i ; r := p ; if isprime(p) then s := p ; for i from 1 to o do r := nextprime(r) ; s := s+r ; end do: r := p ; for i from 1 to o do r := prevprime(r) ; s := s+r ; end do: s := s/(2*o+1) ; if s = p then true; else false; end if; else false; end if; end proc:
isA160920 := proc(p) isBalPr(p,2) and isBalPr(p,3) and isBalPr(p,4) ; end proc:
for i from 10 do p := ithprime(i) ; if isA160920(p) then printf("%d,\n",p); end if; end do: # R. J. Mathar, Dec 15 2010

PrimeNext[n_]:=Module[{k},k=n+1;While[ !PrimeQ[k],k++ ];k];PrimePrev[n_]:=Module[{k},k=n-1;While[ !PrimeQ[k],k-- ];k];lst={};Do[p=Prime[n];a1=PrimePrev[p];a2=PrimePrev[a1];a3=PrimePrev[a2];a4=PrimePrev[a3];a5=PrimePrev[a4];b1=PrimeNext[p];b2=PrimeNext[b1];b3=PrimeNext[b2];b4=PrimeNext[b3];b5=PrimeNext[b4];If[(a1+a2+a3+a4+b1+b2+b3+b4)/8==p&&(a1+a2+a3+b1+b2+b3)/6==p&&(a1+a2+b1+b2)/4==p,AppendTo[lst,p]],{n,2*9!}];lst

A267028 P(n,k) is an array read by rows, with n > 0 and k=1..5, where row n gives the chain of 5 consecutive primes {p(i), p(i+1), p(i+2), p(i+3), p(i+4)} having the symmetrical property p(i) + p(i+4) = p(i+1) + p(i+3) = 2*p(i+2) for some index i.

Original entry on oeis.org

18713, 18719, 18731, 18743, 18749, 25603, 25609, 25621, 25633, 25639, 28051, 28057, 28069, 28081, 28087, 30029, 30047, 30059, 30071, 30089, 31033, 31039, 31051, 31063, 31069, 44711, 44729, 44741, 44753, 44771, 76883, 76907, 76913, 76919, 76943

Offset: 1

Michel Lagneau, Feb 23 2016

a(3 + 5*(n-1)) = A051795(n).
The immediate objective of the sequence is to examine symmetrical properties in the array P(n,k). It is interesting to note that the results with the dimension 5 are generalizable to the dimensions 7, 9, ...
Notation:
We introduce the following function S(i,j) where row i is defined by {P(i,k)} and row j is defined by {P(j,k)}, k = 1..5. Let S(i, j) = 1 if P(i,1) + P(j,5) = P(i,2) + P(j,4) = P(i,3) + P(j,3), otherwise 0.
Conjecture:
For each integer n, there exists an infinite sequence of integers b(n,m), m = 1, 2, ... such that S(n, b(n,m)) = 1.
The following table gives the first values b(n,m).
Notation in the table: "PS" = primitive sequence.
+----+------------------------------------------------+-----------+
|  n |      sequences b(n,m), m=1,2,... of index      |included in|
+----+------------------------------------------------+-----------+
|  1 |  1, 2, 3, 5, 8, 9, 10, 12, 15, 16, 17, 18, ... |     PS    |
|  2 |  2, 3, 5, 8, 9, 10, 12, 15, 16, 17, 18, 19, ...|  {b(1,m)} |
|  3 |  3, 5, 8, 9, 10, 12, 15, 16, 17, 18, 19, ...   |  {b(1,m)} |
|  4 |  4, 6, 11, 13, 14, 21, 28, 35, 39, 57, 59, ... |     PS    |
|  5 |  5, 8, 9, 10, 12, 15, 16, 17, 18, 19, 22, ...  |  {b(1,m)} |
|  6 |  6, 11, 13, 14, 21, 35, 39, 57, 59, 63, 67, ...|  {b(4,m)} |
|  7 |  7, 30, 52, 55, 73, 74, 115, 159, 177, 183, ...|     PS    |
|  8 |  8, 9, 10, 12, 15, 16, 17, 18, 19, 22, 23, ... |  {b(1,m)} |
|  9 |  9, 10, 12, 15, 16, 17, 18, 19, 22, 23, 24, ...|  {b(1,m)} |
| 10 | 10, 12, 15, 16, 17, 18, 19, 22, 23, 24, 26, ...|  {b(1,m)} |
| 11 | 11, 13, 14, 21, 28, 35, 39, 57, 59, 63, 67, ...|  {b(4,m)} |
| 12 | 12, 15, 16, 17, 18, 19, 22, 23, 24, 26, 27, ...|  {b(1,m)} |
| 13 | 13, 14, 21, 28, 35, 39, 57, 59, 63, 67, 70, ...|  {b(4,m)} |
| .. | ...                                            |     ...   |
| 20 | 20, 43, 56, 96, 113, 131, 135, 156, 196, ...   |     PS    |
| 25 | 21, 33, 37, 38, 40, 47, 48, 65, 76, 79, 83, ...|     PS    |
...
Example: S(7, 30) = 1.
We observe primitive sequences {b(n,m)} for n = {1, 4, 7, 20, 25, ...}.
(A primitive sequence is a sequence which is not included in another.)
Properties:
(1) S(i, i)= 1 for all i;
(2) S(i, j) = 1 => S(j, i) = 1;
(3) S(i, j) = 1 and S(j, L) = 1 => S(i, L) = 1.
Example:
For n = 1, {P(1,k)} = {18713, 18719, 18731, 18743, 18749};
we choose, for instance, b(1,2) = 3 => for n = 3, {C(3,k)} = {28051, 28057, 28069, 28081, 28087};
S(1,3) = 1 because 18713 + 28087 = 18719 + 28081 = 18731 + 28069 = 18743 + 28057 = 18749 + 28051 = 46800.
In order to find the index L for satisfying the property (3), we choose, for instance, the index b(3,2) = 8 => for n = 8, {P(8,k)} = {97423, 97429, 97441, 97453, 97459} and S(3, 8) = 1 because 28051 + 97459 = 28057 + 97453 = 28069 + 97441 = 28081 + 97429 = 28087 + 97423 = 125510.
Conclusion: S(1, 3) = 1 and S(3, 8) = 1 => S(1, 8) = 1 with 18713 + 97459 = 18719 + 97453 = 18731 + 97441 =  18743 + 97429 = 18749 + 97423 = 116172.

			The first row is [18713, 18719, 18731, 18743, 18749] because 18713 + 18749 = 18719 + 18743 = 2*18731 = 37462.
The array starts with:
  [18713, 18719, 18731, 18743, 18749]
  [25603, 25609, 25621, 25633, 25639]
  [28051, 28057, 28069, 28081, 28087]
  ...

Cf. A051795, A055380, A055382.

U:=array(1..50,1..5):W:=array(1..2):kk:=0:
for n from 4 to 10000 do:
   for m from 2 by -1 to 1 do:
      q:=ithprime(n-m)+ithprime(n+m):W[m]:=q:
    od:
    if W[1]=W[2] and W[1]=2*ithprime(n) then
    kk:=kk+1:U[kk,1]:=ithprime(n-2):
    U[kk,2]:=ithprime(n-1):U[kk,3]:=ithprime(n):
    U[kk,4]:=ithprime(n+1):U[kk,5]:=ithprime(n+2):
    else fi:od:print(U):
    for i from 1 to kk do:
     for j from i+1 to kk do:
      s1:=U[i,1]+U[j,5]:
      s2:=U[i,2]+U[j,4]:
      s3:=U[i,3]+U[j,3]:
      s4:=U[i,4]+U[j,2]:
      s5:=U[i,5]+U[j,1]:
     if s1=s2 and s2=s3 and s3=s4 and s4=s5
     then
     printf("%d %d \n",i,j):
     else fi:
     od:
  od:

cp's OEIS Frontend

A096710 Quadruply balanced primes: primes which are averages of their immediate neighbor primes, their second neighbor primes, their third neighbor primes and their fourth neighbor primes.

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A126554 Arithmetic mean of two consecutive balanced primes (of order one).

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A082312 Half the difference between start and center prime of the smallest [2n+1]-balanced prime set (A082080).

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PARI

A160920 Primes which are at the same time balanced primes of order 2, 3 and 4.

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GAP

Maple

Mathematica

A267028 P(n,k) is an array read by rows, with n > 0 and k=1..5, where row n gives the chain of 5 consecutive primes {p(i), p(i+1), p(i+2), p(i+3), p(i+4)} having the symmetrical property p(i) + p(i+4) = p(i+1) + p(i+3) = 2*p(i+2) for some index i.

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Maple