A096706 -id:A096706 - cp's OEIS frontend

A096693 Balance index of each prime.

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

A096707 Balanced primes (A090403) of index 3.

Original entry on oeis.org

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

A096705 Balanced primes of index 1.

Original entry on oeis.org

5, 17, 29, 37, 71, 79, 89, 137, 149, 151, 179, 193, 227, 229, 257, 281, 359, 373, 383, 419, 421, 433, 487, 491, 563, 577, 593, 631, 643, 653, 659, 677, 701, 733, 757, 823, 877, 947, 953, 983, 991, 1013, 1021, 1087, 1091, 1103, 1123, 1171, 1193, 1217, 1223

Offset: 1

Robert G. Wilson v, Jun 28 2004

nonn

			17 is a member because 17 = (7 + 11 + 13 + 17 + 19 + 23 + 29)/7 only.

Robert Price, Table of n, a(n) for n = 1..4957

Cf. A096693, A096695, A096706, A096707, A096708, A096709.

g[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[ # ] == 1 &]]

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

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A096693 Balance index of each prime.

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A096707 Balanced primes (A090403) of index 3.

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A096705 Balanced primes of index 1.

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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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