A006562 -id:A006562 - cp's OEIS frontend

A119403 Primes p=prime(i) of level (1,10), i.e., such that A118534(i)=prime(i-10).

745757, 1103639, 1583369, 1895359, 2124049, 3327419, 4234537, 4437779, 5071973, 6287647, 7702573, 8470927, 8675923, 9493151, 9750079, 10868203, 11213843, 14244173, 14796253, 14978893, 15611909, 16489273, 17528681, 18280771, 19125163, 19403831, 19631411, 21975167

Offset: 1

Rémi Eismann and Fabien Sibenaler, Jul 25 2006

nonn

This subsequence of A125830 and of A162174 gives primes of level (1,10): If prime(i) has level 1 in A117563 and 2*prime(i) - prime(i+1) = prime(i-k), then we say that prime(i) has level (1,k).

			prime(353166) - prime(353165) = 5072057 - 5071973 = 5071973 - 5071889 = prime(353165) - prime(353165-10) and prime(353165) has level 1 in A117563, so prime(353165)=5071973 has level (1,10).

Fabien Sibenaler, Table of n, a(n) for n = 1..10000

Cf. A117078, A117563, A006562 (primes of level (1,1)), A117876, A118464, A118467.

lista(nn) = my(c=11, v=primes(11)); forprime(p=37, nn, if(2*v[c]-p==v[c=c%11+1], print1(precprime(p-1), ", ")); v[c]=p); \\ Jinyuan Wang, Jun 18 2021

More terms from Fabien Sibenaler, Oct 20 2006

Definition and comment reworded following suggestions from the authors. - M. F. Hasler, Nov 30 2009

A119404 Primes p=prime(i) of level (1,9), i.e., such that A118534(i)=prime(i-9).

Original entry on oeis.org

678659, 855739, 1403981, 2366543, 2744783, 2830657, 3027539, 3317033, 4525909, 4676851, 5341463, 5819563, 7087123, 7181897, 8815663, 9324257, 9878929, 9976937, 10403251, 10440641, 10447457, 10766411, 10787377, 11829151, 11881957, 12539389, 14026433, 14087179

Offset: 1

Rémi Eismann and Fabien Sibenaler, Jul 25 2006

nonn

This subsequence of A125830 and of A162174 gives primes of level (1,9): If prime(i) has level 1 in A117563 and 2*prime(i) - prime(i+1) = prime(i-k), then we say that prime(i) has level (1,k).

			prime(780815) - prime(780814) = 11882071 - 11881957 = 11881957 - 11881843 = prime(780814) - prime(780814-9) and prime(780814) has level 1 in A117563, so prime(780814)=11881957 has level (1,9).

Fabien Sibenaler, Table of n, a(n) for n = 1..10000

Cf. A117078, A117563, A006562 (primes of level (1,1)), A117876, A118464, A118467.

Definition and comment reworded following suggestions from the authors. - M. F. Hasler, Nov 30 2009

A359440 A measure of the extent of reflective symmetry in the pattern of primes around each prime gap: a(n) is the largest k such that prime(n-j) + prime(n+1+j) has the same value for each j in 0..k.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 1, 0, 0, 4, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0

Offset: 1

Alexandre Herrera, Jan 01 2023

nonn

If the prime gaps above and below a prime p have the same length, p is called a balanced prime (see A006562). Likewise, if the prime gaps above and below the n-th prime gap have the same length, this gap might be called a balanced prime gap. These gaps correspond to nonzero terms a(n). Similarly, if a(n) >= 2, the n-th prime gap is the equivalent of a doubly balanced prime (A051795), and so on. - Peter Munn, Jan 08 2023

			For n = 1, prime(1) + prime(2) = 2 + 3 = 5; "prime(0)" does not exist, so a(1) = 0.
For n = 4:
  j = 0:  prime(4) + prime(5) =  7 + 11 = 18;
  j = 1:  prime(3) + prime(6) =  5 + 13 = 18;
  j = 2:  prime(2) + prime(7) =  3 + 17 = 20 != 18, so a(4) = 1.
For n = 5:
  j = 0:  prime(5) + prime(6) = 11 + 13 = 24;
  j = 1:  prime(4) + prime(7) =  7 + 17 = 24;
  j = 2:  prime(3) + prime(8) =  5 + 19 = 24;
  j = 3:  prime(2) + prime(9) =  3 + 23 = 26 != 24, so a(5) = 2.

Cf. A000040, A006562, A051795, A055381, A081235.

import sympy
offset = 1
N = 100
l = []
for n in range(offset,N+1):
    j = 0
    first_sum = sympy.prime(n-j)+sympy.prime(n+j+1)
    while (n-j) > 1:
        j += 1
        sum = sympy.prime(n-j)+sympy.prime(n+j+1)
        if sum != first_sum:
            break
    l.append(max(0,j-1))
print(l)

a(n) = min( {n-1} U {k : 0 <= k <= n-2 and prime(n-k-1) + prime(n+k+2) <> prime(n) + prime(n+1)} ). - Peter Munn, Jan 08 2023

Introductory phrase added to name by Peter Munn, Jan 08 2023

A053709 Prime balanced factorials: numbers k such that k! is the mean of its 2 closest primes.

Original entry on oeis.org

3, 5, 10, 21, 171, 190, 348, 1638, 3329

Offset: 1

Labos Elemer, Feb 10 2000

Also, the integers k such that A033932(k) = A033933(k).
k! is an interprime, i.e., the average of two successive primes.
The difference between k! and any of its two closest primes must be 1 or exceed k. - Franklin T. Adams-Watters
Larger terms may involve probable primes. - Hans Havermann, Aug 14 2014

			For the 1st term, 3! is in the middle between its closest prime neighbors 5 and 7.
For the 2nd term, 5! is in the middle between its closest prime neighbors 113 and 127.
From _Jon E. Schoenfield_, Jan 14 2022: (Start)
In the table below, k = a(n), k! - d and k! + d are the two closest primes to k!, and d = A033932(k) = A033933(k) = A053711(n):
.
  n     k     d
  -  ----  ----
  1     3     1
  2     5     7
  3    10    11
  4    21    31
  5   171   397
  6   190   409
  7   348  1657
  8  1638  2131
  9  3329  7607
(End)

Cf. A053711 (distances), A033932, A033933, A006990, A037151, A006562, A053710, A075275.

Cf. A075409 (smallest m such that n!-m and n!+m are both primes).

for n from 3 to 200 do j := n!-prevprime(n!): if not isprime(n!+j) then next fi: i := 1: while not isprime(n!+i) and (i<=j) do i := i+2 od: if i=j then print(n):fi:od:

PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k] Do[ a = n!; If[2a == PrevPrim[a] + NextPrim[a], Print[n]], {n, 3, 415}]

a(5)-a(6) from Jud McCranie, Jul 04 2000
a(7) from Robert G. Wilson v, Sep 17 2002
a(8) from Donovan Johnson, Mar 23 2008
a(9) from Hans Havermann, Aug 14 2014

A075540 Integers that are the average of three successive primes.

Original entry on oeis.org

5, 53, 157, 173, 211, 257, 263, 373, 511, 537, 563, 593, 607, 653, 733, 947, 977, 999, 1073, 1103, 1123, 1187, 1223, 1239, 1367, 1461, 1501, 1511, 1541, 1747, 1753, 1763, 1773, 1899, 1907, 1917, 2071, 2181, 2287, 2401, 2409, 2417, 2449, 2677, 2903, 2963

Offset: 1

Zak Seidov, Sep 21 2002

nonn

Not every three successive primes have an integer average. The integer averages are in the sequence.

Not all of these 3-averages are prime: the prime 3-averages are in A006562 (balanced primes). There are surprisingly many prime 3-averages: among the first 10000 terms of the sequence there are 2417 primes. Indices i(n) of first prime in sequence of three primes with integer average are in A075541, for prime 3-averages i(n) are in A064113. Interprimes (s-averages with s=2) are all composite, see A024675. (Edited by Zak Seidov, Sep 01 2015 )

			a(1) = 5 = (1/3)(3+5+7), first integer average of three successive primes; next is: a(2) = 53 = (1/3)(47 + 53 + 59); up to n=8 all terms are prime; while a(9) = 511 = (1/3)( 503 + 509 + 521) is the first nonprime 3-average: 511=7*73.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A006562, A024675, A075541, A064113.

Cf. A102655.

a075540 n = a075540_list !! (n-1)
a075540_list = map fst $ filter ((== 0) . snd) $
   zipWith3 (\x y z -> divMod (x + y + z) 3)
            a000040_list (tail a000040_list) (drop 2 a000040_list)
-- Reinhard Zumkeller, Jan 20 2012

N:= 10^4: # to get all terms using primes <= N
Primes:= select(isprime,[2,seq(2*i+1, i=1..(N-1)/2)]):
select(type,(Primes[1..-3] + Primes[2..-2] + Primes[3..-1])/3,integer); # Robert Israel, Sep 01 2015

Select[MovingAverage[Prime[Range[500]],3],IntegerQ] (* Harvey P. Dale, Aug 10 2012 *)

a(n) = (1/3) (p(i)+p(i+1)+p(i+2)), for some i(n).

Comment and example edited, inefficient Mma removed by Zak Seidov, Sep 01 2015

A081415 Triply balanced primes: primes which are averages of both their immediate neighbor, their second neighbors and their third neighbors.

Original entry on oeis.org

683783, 1056317, 1100261, 2241709, 2815301, 4746359, 10009049, 12003209, 13810981, 14907649, 15403009, 15730067, 16595081, 17518201, 19755301, 20378327, 21006487, 21574453, 21579983, 22237121, 22625179, 25876901, 26018791, 26354201, 27188141, 28469461

Offset: 1

Labos Elemer, Apr 02 2003

nonn

Equivalently, primes which are balanced primes of orders 1, 2, and 3. - Muniru A Asiru, Apr 08 2018

Numbers m such that A346399(m) is odd and >= 7. - Ya-Ping Lu, May 11 2024

			p = 683383: 683747 + ... + p + ... + 683819 = 7p; 683759 + ... + p + ... + 683807 = 5p; 683777 + p + 683789 = 3p.

Jud McCranie, Table of n, a(n) for n = 1..1000
Wikipedia, Balanced prime

Cf. A006562, A051795, A055380, A096710, A346399.

P:=Filtered([1,3..3*10^7+1],IsPrime);;
a:=Intersection(List([1,2,3],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

a = {}; Do[p = 2Prime[n]; If[p == Prime[n - 1] + Prime[n + 1] && p == Prime[n - 2] + Prime[n + 2] && p == Prime[n - 3] + Prime[n + 3], Print[p / 2]; AppendTo[a, p / 2]], {n, 5, 1100000}]; a (* Robert G. Wilson v, Jun 28 2004 *)
Transpose[Select[Partition[Prime[Range[1620000]],7,1],(#[[1]]+#[[7]])/2 == (#[[2]]+#[[6]])/2==(#[[3]]+#[[5]])/2==#[[4]]&]][[4]] (* Harvey P. Dale, Sep 13 2013 *)

from sympy import nextprime; p, q, r, s, t, u, v = 2, 3, 5, 7, 11, 13, 17
while v < 29000000:
    if p + v == q + u == r + t == 2*s: print(s, end = ', ')
    p, q, r, s, t, u, v = q, r, s, t, u, v, nextprime(v) # Ya-Ping Lu, May 11 2024

A082080 Smallest balanced prime of order n.

Original entry on oeis.org

2, 5, 79, 17, 491, 53, 71, 29, 37, 983, 5503, 173, 157, 353, 5297, 263, 179, 383, 137, 2939, 2083, 751, 353, 5501, 1523, 149, 4561, 1259, 397, 787, 8803, 8803, 607, 227, 3671, 17443, 57097, 3607, 23671, 12539, 1217, 11087, 1087, 21407, 19759, 953

Offset: 0

Labos Elemer, Apr 08 2003

nonn

Or, smallest (2n+1)-balanced prime number.

Prime(k) is a balanced prime of order n if it is the average of the 2n+1 primes from prime(k-n) to prime(k+n).

			a(1) = 5 = (3 + 5 + 7)/3 = 15/3.
a(5) = 53 = (31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73)/11 = 583/11.
a(6) = 71 = (43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101)/13 = 923/13.

Giovanni Resta, Table of n, a(n) for n = 0..10000 (first 1375 terms from Robert G. Wilson v)

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

Cf. A006562, A051795, A081415, A096710, A055380, A082312, A075540, A054800.

f[n_] := Block[{p = Prime@ Range[2n +1]}, While[ Total[p] != (2n +1) p[[n +1]], p = Join[Rest@ p, {NextPrime[ p[[-1]]] }]]; p[[n +1]]]; Array[f, 46, 0] (* Robert G. Wilson v, Jun 21 2004 and modified Apr 11 2017 *)

for(n=0, 50, 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", ");f=1;break)); if(!f, print1("0, ")))

Corrected and extended by Ralf Stephan, Apr 09 2003

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.

Original entry on oeis.org

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

A178609 Largest k < n such that prime(n-k) + prime(n+k) = 2*prime(n).

Original entry on oeis.org

0, 0, 1, 0, 3, 2, 2, 0, 0, 5, 3, 6, 4, 0, 0, 7, 7, 4, 8, 10, 0, 0, 7, 4, 11, 6, 2, 2, 0, 0, 13, 9, 10, 12, 0, 2, 16, 0, 6, 12, 13, 4, 19, 17, 15, 0, 18, 0, 0, 0, 11, 0, 0, 3, 1, 1, 0, 0, 6, 0, 0, 0, 27, 13, 0, 0, 17, 5, 29, 23, 26, 20, 26, 11, 7, 21, 20, 15, 19, 34, 21, 2, 21, 11, 10, 10, 10, 27, 3, 0, 0, 5, 32, 2, 0, 0, 0, 26, 0, 33

Offset: 1

Juri-Stepan Gerasimov, Dec 24 2010

The plot is very interesting.

			a(3)=1 because 5=prime(3)=(prime(3-1)+prime(3+1))/2=(3+7)/2.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A006562 (balanced primes), A178670 (number of k), A178698 (composite case), A179835 (smallest k).

a178609 n = head [k | k <- [n - 1, n - 2 .. 0], let p2 = 2 * a000040 n,
                      a000040 (n - k) + a000040 (n + k) == p2]
-- Reinhard Zumkeller, Jan 30 2014

Table[k=n-1; While[Prime[n-k]+Prime[n+k] != 2*Prime[n], k--]; k, {n,100}]

def A178609(n):
    return next(k for k in range(n)[::-1] if nth_prime(n-k)+nth_prime(n+k) == 2*nth_prime(n))
# D. S. McNeil, Dec 29 2010

a(A178953(n)) = 0.

Extended and corrected by T. D. Noe, Dec 28 2010

A126555 Primes in A126554.

Original entry on oeis.org

29, 1439, 4211, 7703, 12907, 14957, 16703, 20947, 29221, 31189, 33053, 36749, 44531, 51437, 90247, 115547, 124783, 127163, 133337, 144511, 146449, 151339, 157133, 166219, 169241, 180233, 181031, 185123, 197383, 223367, 225287, 240347

Offset: 1

Artur Jasinski, Dec 27 2006

nonn

Primes that are the arithmetic mean of two consecutive balanced primes (of order one); primes of the form (A006562(k)+A006562(k+1))/2.

Might be called prime interprimes of order two.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A126554, A006562.

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, 1,10000}];Do[If[PrimeQ[(a[[k + 1]] + a[[k]])/2], AppendTo[b, (a[[k + 1]] + a[[k]])/2]], {k, 1, Length[a] - 1}]; b

{m=250000;a=0;p=2;q=3;r=5;while(r<=m,if((p+r)/2==q,if(a>0,if(isprime(b=(a+q)/2),print1(b,",")));a=q);p=q;q=r;r=nextprime(r+1))} \\ Klaus Brockhaus, Jan 05 2007

Edited and extended by Klaus Brockhaus, Jan 05 2007

cp's OEIS Frontend

A119403 Primes p=prime(i) of level (1,10), i.e., such that A118534(i)=prime(i-10).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A119404 Primes p=prime(i) of level (1,9), i.e., such that A118534(i)=prime(i-9).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A359440 A measure of the extent of reflective symmetry in the pattern of primes around each prime gap: a(n) is the largest k such that prime(n-j) + prime(n+1+j) has the same value for each j in 0..k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

Formula

Extensions

A053709 Prime balanced factorials: numbers k such that k! is the mean of its 2 closest primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

A075540 Integers that are the average of three successive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

Extensions

A081415 Triply balanced primes: primes which are averages of both their immediate neighbor, their second neighbors and their third neighbors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Mathematica

Python

A082080 Smallest balanced prime of order n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI