A090403 -id:A090403 - cp's OEIS frontend

A096707 Balanced primes (A090403) of index 3.

53, 607, 977, 1289, 2083, 2351, 4013, 5563, 8803, 10657, 11117, 12583, 14747, 16433, 18731, 22067, 22699, 28477, 32833, 39227, 39749, 41957, 44357, 46229, 46643, 50053, 50123, 51869, 53617, 54469, 56167, 63377, 63527, 66797, 74729, 75217, 76597, 77023, 93997

Offset: 1

Robert G. Wilson v, Jun 28 2004

nonn

			607 is a member because 607 = (601 + 607 + 613)/3 =
(593 + 599 + 601 + 607 + 613 + 617 + 619)/7 = (401 + ... + 607 + ... + 823)/65.

Robert Price, Table of n, a(n) for n = 1..85

Cf. A096693, A096695, A096705, A096706, A096708, A096709.

f[n_] := Block[{c = 0, k = 1, p = Prime[n], s = Plus @@ Table[ Prime[i], {i, n - 1, n + 1}]}, While[k != n - 1, If[s == (2k + 1)p, c++ ]; k++; s = s + Prime[n - k] + Prime[n + k]]; c]; Prime[ Select[ Range[2, 250], f[ # ] == 3 &]]

a(37)-a(39) from Robert Price, Nov 29 2023

A096706 Balanced primes (A090403) of index 2.

Original entry on oeis.org

211, 263, 349, 397, 409, 439, 709, 751, 787, 827, 1153, 1187, 1259, 1487, 1523, 1531, 2281, 2287, 2347, 2621, 3037, 3109, 3313, 3329, 3539, 3673, 4357, 4397, 4493, 4951, 4969, 4987, 5189, 5303, 5347, 5857, 6323, 6337, 7583, 7907, 7933, 8429, 8713, 8821, 8951

Offset: 1

Robert G. Wilson v, Jun 28 2004

nonn

			263 is a member because 263 = (257 + 263 + 269)/3
= (179 + 181 + 191 + 193 + 197 + 199 + 211 + 223 + 227 + 229 + 233 + 239 + 241 + 251 + 257 + 263 + 269 + 271 + 277 + 281 + 283 + 293 + 307 + 311 + 313 + 317 + 331 + 337 + 347 + 349 + 353)/31.

Robert Price, Table of n, a(n) for n = 1..753

Cf. A096693, A096695, A096705, A096707, A096708, A096709.

f[n_] := Block[{c = 0, k = 1, p = Prime[n], s = Plus @@ Table[ Prime[i], {i, n - 1, n + 1}]}, While[k != n - 1, If[s == (2k + 1)p, c++ ]; k++; s = s + Prime[n - k] + Prime[n + k]]; c]; Prime[ Select[ Range[2, 250], f[ # ] == 2 &]]

a(45) from Robert Price, Nov 29 2023

A096709 Balanced primes (A090403) of index 5.

Original entry on oeis.org

173, 124991, 232607, 491423, 701489, 1356337, 2455681, 3128803, 5218607, 9459683, 10563461, 13228247, 14606029, 16282921, 18216137, 20378273, 21622201, 35201909, 36549169, 38638969, 39246689, 42074017, 43048039, 48961859

Offset: 1

Robert G. Wilson v, Jun 28 2004

nonn

			124991 is a member because 124991 = (124673 + ... + 125329)/59
= (124543 + ... + 125423)/75 = (124193 + ... + 125777)/137 = (124133 + ... + 125887)/151
= (123931 + ... + 126031)/181.

Cf. A096693, A096695, A096705, A096706, A096707, A096708.

f[n_] := Block[{c = 0, k = 1, p = Prime[n], s = Plus @@ Table[ Prime[i], {i, n - 1, n + 1}]}, While[k != n - 1, If[s == (2k + 1)p, c++ ]; k++; s = s + Prime[n - k] + Prime[n + k]]; c]; Prime[ Select[ Range[2, 25797], f[ # ] == 5 &]]

a(6)-a(24) from Donovan Johnson, Apr 09 2010

A096695 Least balanced prime (A090403) of index n (A096693).

Original entry on oeis.org

2, 5, 211, 53, 157, 173, 304517

Offset: 0

Robert G. Wilson v, Jun 26 2004

nonn

			a(1) = 5 because 5 = (3 + 5 + 7)/3.
a(2) = 211 because 211 = (199 + 211 + 223)/3
= (179 + 181 + 191 + 193 + 197 + 199 + 211 + 223 + 227 + 229 + 233 + 239 + 241)/13.
a(3) = 53 because 53 = (47 + 53 + 59)/3 = (41 + 43 + 47 + 53 + 59 + 61 + 67)/7
= (31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73)/11.
a(4) = 157 because 157 = (151 + 157 + 163)/3 = (139 + 149 + 151 + 157 + 163 + 167 + 173)/7
= (131 + 137 + 139 + 149 + 151 + 157 + 163 + 167 + 173 + 179 + 181)/11
= (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.
a(5) = 173 because 173 = (167 + 173 + 179)/3 = (157 + 163 + 167 + 173 + 179 + 181 + 191)/7
= (131 + 137 + 139 + 149 + 151 + 157 + 163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211 + 223)/17
= (109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 + 163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211 + 223 + 227 + 229 + 233)/23
= (107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 + 163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211 + 223 + 227 + 229 + 233 + 239)/25.
a(6) = 304517 because 304517 = (304511 + 304517 + 304523)/3
= (304489 + 304501 + 304511 + 304517 + 304523 + 304537 + 304541)/7
= (304481 + 304489 + 304501 + 304511 + 304517 + 304523 + 304537 + 304541 + 304553)/9
= (303691 + ... + 304517 + ... + 305339)/135 = (303649 + ... + 304517 + ... + 305369)/143

Cf. A096693, A096705, A096706, A096707, A096708, A096709.

f[n_] := Block[{c = 0, k = 1, p = Prime[n], s = Plus @@ Table[ Prime[i], {i, n - 1, n + 1}]}, While[k != n - 1, If[s == (2k + 1)p, c++ ]; k++; s = s + Prime[n - k] + Prime[n + k]]; c]; t = Table[0, {15}]; Do[a = f[n]; If[a < 50 && t[[a + 1]] == 0, t[[a + 1]] = Prime[n]; Print[a + 1, " = ", Prime[n]]], {n, 32000}]; t

A096708 Balanced primes (A090403) of index 4.

Original entry on oeis.org

157, 353, 8233, 23893, 26183, 30197, 63697, 118831, 131041, 150203, 152213, 167033, 198013, 293087, 341303, 383983, 494051, 494723, 534007, 551569, 601949, 603541, 629203, 666697, 671287, 679417, 688907, 768203, 787207, 796867, 826039

Offset: 1

Robert G. Wilson v, Jun 28 2004

nonn

			353 is a member because 353 = (281 + ...353 + ... + 421)/23
= (271 + .. + 353 + ... + 433)/27 = (241 + ... + 353 + ... + 461)/37 = (227 + ... + 353 + ... + 487)/45.

Cf. A096693, A096695, A096705, A096706, A096707, A096709.

f[n_] := Block[{c = 0, k = 1, p = Prime[n], s = Plus @@ Table[ Prime[i], {i, n - 1, n + 1}]}, While[k != n - 1, If[s == (2k + 1)p, c++ ]; k++; s = s + Prime[n - k] + Prime[n + k]]; c]; Prime[ Select[ Range[2, 25797], f[ # ] == 4 &]]

a(17)-a(31) from Donovan Johnson, Apr 09 2010

A096711 Number of balanced primes (A090403) less than 10^n.

Original entry on oeis.org

1, 8, 57, 308, 1989, 13161, 94939

Offset: 1

Robert G. Wilson v, Jul 03 2004

nonn

The average number of balanced primes, p_n, seems to reach a maximum at the 85th prime, 439, of 32 balanced primes.

A096711(n)/A006880(n) begins 25%, 32%, 33.93%, 25.06%, 20.74%

Cf. A090403.

f[n_] := f[n] = Block[{c = 0, k = 1, p = Prime[n], s = Plus @@ Table[Prime[i], {i, n - 1, n + 1}]}, While[k != n - 1, If[s == (2k + 1)p, Return[1]]; k++; s = s + Prime[n - k] + Prime[n + k]]; 0]; f[1] = 0; Do[ f[n], {n, 10000}]; s = Prime[ Select[ Range[ 10000], f[ # ] == 1 &]]; Table[ Length[ Select[s, # < 10^n &]], {n, 5}]

a(6)-a(7) from Donovan Johnson, Apr 09 2010

A096694 Lesser of twin balanced primes (A090403).

Original entry on oeis.org

149, 227, 419, 1997, 3329, 3671, 5501, 6449, 13691, 15887, 21647, 22481, 26711, 27749, 31247, 32411, 32831, 37547, 39227, 41759, 44027, 47777, 49121, 50261, 53231, 54539, 54917, 55217, 64877, 69149, 71411, 74717, 90821, 93239, 107069

Offset: 1

Robert G. Wilson v, Jun 28 2004

nonn

Cf. A090403.

f[n_] := f[n] = Block[{c = 0, k = 1, p = Prime[n], s = Plus @@ Table[ Prime[i], {i, n - 1, n + 1}]}, While[k != n - 1, If[s == (2k + 1)p, c++ ]; k++; s = s + Prime[n - k] + Prime[n + k]]; c]; p = Prime[ Select[ Range[2, 22456], f[ # ] != 0 &]]; Transpose[ Select[ Partition[p, 2, 1], #[[1]] + 2 == #[[2]] &]][[1]]

A096693 Balance index of each prime.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 4, 0, 0, 5, 1, 0, 0, 1, 0, 0, 2, 0, 1, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 1, 0, 1, 0, 1, 0, 2, 0, 2, 1, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Robert G. Wilson v, Jun 26 2004

nonn

a(n) = the number of values of k for which the n-th prime is equal to the arithmetic average of the k primes above and below it.

The average of the first n balance indexes appears to reach a global maximum of 0.588 when n = 85, (prime(85) = 439).

			a(3) = 1 because the third prime, 5, equals (3 + 7)/2.
a(16) = 3 because the sixteenth prime, 53, equals (47 + 59)/2 = (41 + 43 + 47 + 59 + 61 + 67)/6 = (31 + 37 + 41 + 43 + 47 + 59 + 61 + 67 + 71 + 73)/10.

C. H. Gribble, Table of n, a(n) for n=1,..., 10000.

Cf. A090403, A096695, A096705, A096706, A096707, A096708, A096709, A096711.

f[n_] := Block[{c = 0, k = 1, p = Prime[n], s = Plus @@ Table[ Prime[i], {i, n - 1, n + 1}]}, While[k != n - 1, If[s == (2k + 1)p, c++ ]; k++; s = s + Prime[n - k] + Prime[n + k]]; c]; Table[ f[n], {n, 105}]

b-file generator: {max_n = 10^4; for (n = 1, max_n, c = 0; k = 1; p = prime(n); s = p; while (k < n, s = s + prime(n - k) + prime(n + k); if (s == (2 * k + 1) * p, c++); k++;); print(n " " c);) ;}

Corrected and edited by Christopher Hunt Gribble, Apr 06 2010

A089180 a(n) is the smallest number m such that d(m) = d(m+1) = ... = d(m+n), where d(k) = prime(k+1) - prime(k) (A001223).

Original entry on oeis.org

2, 54, 654926, 6904737

Offset: 1

Farideh Firoozbakht, Dec 07 2003

a(5) is greater than 105000000.

The a(n)-th prime is the smallest start of n+2 consecutive primes in arithmetic progression. - Jens Kruse Andersen, Jun 14 2014

			a(3) = 659426 because d(659426) = d(659426+1) = d(659426+2) = d(6594286+3) or 9843019, 9843049, 9843079, 9843109, 9843139 are five consecutive primes with same difference and prime(659426) = 9843019 is the smallest prime number with this property.
Also a(4) = 6904737 because d(6904737) = d(6904737+1) = ... = d(6904737+4) or 121174811, 121174841, 121174871, 121174901, 121174931, 121174961 are six consecutive primes with same difference and prime(6904737) = 121174811 is the smallest prime number with this property.

J. K. Andersen, The minimal CPAP-k.
L. J. Lander and T. R. Parkin, Consecutive primes in arithmetic progression, Math. Comp. vol. 21 no. 99 (1967) p. 489.
G. W. Polites, Prime Desert n-Tuplets, Amer. Math. Monthly vol. 95 no. 2 (1988) pp. 98-104.

Cf. A001223, A090403.

A000040[a(n)]=A006560(n+2). - R. J. Mathar, Aug 10 2007

a(n) = A000720(A006560(n+2)). - Jens Kruse Andersen, Jun 14 2014

A096696 Consider the n-th prime, p_n, as the beginning of 2k+1 consecutive primes; then a(n) = p_(n+k) a balanced prime of order k, k maximized, or 0 if no such prime exists.

Original entry on oeis.org

0, 5, 29, 37, 0, 0, 0, 0, 0, 149, 53, 0, 53, 71, 137, 227, 0, 0, 89, 79, 0, 0, 0, 0, 179, 0, 0, 173, 173, 0, 0, 419, 0, 157, 0, 157, 173, 0, 173, 0, 263, 0, 0, 0, 0, 211, 229, 0, 353, 397, 0, 0, 353, 359, 409, 577, 0, 353, 383, 353, 0, 0, 0, 0, 0, 0, 349, 349, 0, 0, 0, 397, 373

Offset: 1

Robert G. Wilson v, Jul 02 2004

nonn

a(n) either equals 0 or belongs to A090403.

			a(2) = 5 because beginning with the second prime, 3, there is a run of three prime, (3,5,7) the average and median of which is 5.
a(5) = 0 because there does not exist a run of 2k + 1 primes such that the arithmetic mean and the median are the same.

Cf. A090403.

f[n_] := Block[{k = 1, p = 0}, While[k < 10^4, If[(Plus @@ Table[Prime[i], {i, n, n + 2k}]) == (2k + 1)Prime[n + k], p = Prime[n + k]]; k++ ]; p]; Table[ f[n], {n, 74}]

cp's OEIS Frontend

A096707 Balanced primes (A090403) of index 3.

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A096706 Balanced primes (A090403) of index 2.

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A096709 Balanced primes (A090403) of index 5.

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A096695 Least balanced prime (A090403) of index n (A096693).

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A096708 Balanced primes (A090403) of index 4.

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A096711 Number of balanced primes (A090403) less than 10^n.

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A096694 Lesser of twin balanced primes (A090403).

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A096693 Balance index of each prime.

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A089180 a(n) is the smallest number m such that d(m) = d(m+1) = ... = d(m+n), where d(k) = prime(k+1) - prime(k) (A001223).

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