A096700 -id:A096700 - cp's OEIS frontend

A096704 Balanced primes of order twelve.

157, 173, 709, 827, 1999, 2689, 6803, 11351, 11489, 12757, 15277, 33071, 37967, 38449, 46751, 47303, 51599, 53113, 56779, 57269, 59107, 62731, 62743, 62791, 63649, 77023, 79357, 81553, 81649, 81953, 85621, 96377, 108139, 113983, 117839

Offset: 1

Robert G. Wilson v, Jun 26 2004

nonn

			157 is a term because 157 = (97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 + 163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211 + 223 + 227)/25 = 3925/25.

Muniru A Asiru, Table of n, a(n) for n = 1..5000

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

P:=Filtered([1..150000],IsPrime);;
a:=List(Filtered(List([0..12000],k->List([1..25],j->P[j+k])),i->Sum(i)/25=i[13]),m->m[13]); # Muniru A Asiru, Mar 04 2018

Select[Partition[Prime[Range[12000]],25,1],Mean[#]==#[[13]]&][[All,13]] (* Harvey P. Dale, Jun 28 2020 *)

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

Original entry on oeis.org

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

Farideh Firoozbakht, Dec 07 2003

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.

			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.

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

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

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]

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

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

Edited by M. F. Hasler, Oct 21 2012

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

A300365 Balanced primes of order fourteen.

Original entry on oeis.org

5297, 15647, 22073, 22501, 26309, 34721, 43499, 44111, 48809, 57529, 58171, 66797, 69151, 70199, 74551, 76493, 86959, 91297, 93169, 93199, 94343, 102217, 110777, 112289, 113093, 132361, 133493, 135461, 139921, 146021, 155303, 156521, 162557, 163753, 163789

Offset: 1

Muniru A Asiru, Mar 04 2018

nonn

			5297 is a member because 5297 = 5167 + 5171 + 5179 + 5189 + 5197 + 5209 + 5227 + 5231 + 5233 + 5237 + 5261 + 5273 + 5279 + 5281 + 5297 + 5303 + 5309 + 5323 + 5333 + 5347 + 5351 + 5381 + 5387 + 5393 + 5399 + 5407 + 5413 + 5417 + 5419  = 153613/29.

Muniru A Asiru, Table of n, a(n) for n = 1..6600

Cf. Balanced primes of order b: A006562 (b=1), A082077 (b=2), A082078 (b=3), A082079 (b=4), A096697 (b=5), A096698 (b=6), A096699 (b=7), A096700 (b=8), A096701 (b=9), A096702 (b=10), A096703 (b=11), A096704 (b=12), A300364 (b=13) this sequence (b=14).

P:=Filtered([1..200000],IsPrime);;
a:=List(Filtered(List([0..17000],k->List([1..29],j->P[j+k])),i->Sum(i)/29=i[15]),m->m[15]);

Module[{bal=14,nn=16000},Select[Partition[Prime[Range[nn]],2bal+1,1],Mean[#]==#[[bal+1]]&]][[;;,15]] (* Harvey P. Dale, Jul 07 2023 *)

A300364 Balanced primes of order thirteen.

Original entry on oeis.org

353, 2351, 3673, 3863, 4759, 6271, 8539, 8821, 11261, 12073, 17839, 26711, 32797, 33769, 34679, 41357, 53269, 60217, 64879, 64891, 68713, 88493, 90121, 91811, 101347, 101411, 101641, 108139, 108203, 114659, 122029, 123637, 127843, 128237, 130447, 133279

Offset: 1

Muniru A Asiru, Mar 04 2018

nonn

			353 is a member because 353 = 271 + 277 + 281 + 283 + 293 + 307 + 311 + 313 + 317 + 331 + 337 + 347 + 349 + 353 + 359 + 367 + 373 + 379 + 383 + 389 + 397 + 401 + 409 + 419 + 421 + 431 + 433 = 9531/27.

Muniru A Asiru, Table of n, a(n) for n = 1..7200

Cf. Balanced primes of order b: A006562 (b=1), A082077 (b=2), A082078 (b=3), A082079 (b=4), A096697 (b=5), A096698 (b=6), A096699 (b=7), A096700 (b=8), A096701 (b=9), A096702 (b=10), A096703 (b=11), A096704 (b=12), this sequence (b=13), A300365 (b=14).

P:=Filtered([1..150000],IsPrime);;
a:=List(Filtered(List([0..13000],k->List([1..27],j->P[j+k])),i->Sum(i)/27=i[14]),m->m[14]);

A363168 Balanced primes of order 100.

Original entry on oeis.org

27947, 111337, 193283, 197341, 197621, 347063, 809821, 955193, 1029803, 1184269, 1292971, 1609163, 1630859, 1656019, 1752449, 1883381, 1935517, 1969661, 2120221, 2156383, 2238959, 2287133, 2548631, 2592089, 2750903, 2866403, 3165769, 3257941, 3590299, 3889423

Offset: 1

Harvey P. Dale, Jul 07 2023

nonn

A prime p is in this sequence if the sum of the 100 consecutive primes just less than p, plus p, plus the sum of the 100 consecutive primes just greater than p, divided by 201 equals p.

Harvey P. Dale, Table of n, a(n) for n = 1..350 (all terms up to the 4 millionth prime)

Cf. Balanced primes of order b: A006562 (b=1), A082077 (b=2), A082078 (b=3), A082079 (b=4), A096697 (b=5), A096698 (b=6), A096699 (b=7), A096700 (b=8), A096701 (b=9), A096702 (b=10), A096703 (b=11), A096704 (b=12), A300364 (b=13), A300365 (b=14).

Module[{bal=100,nn=300000},Select[Partition[Prime[Range[nn]],2bal+1,1],Mean[#]== #[[bal+1]]&]] [[;;,101]]

cp's OEIS Frontend

A096704 Balanced primes of order twelve.

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

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A082080 Smallest balanced prime of order n.

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A300365 Balanced primes of order fourteen.

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A300364 Balanced primes of order thirteen.

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A363168 Balanced primes of order 100.

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