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Representations as the sum of one or more consecutive primes

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2011 (a prime number) is the sum of eleven consecutive primes ending in 211!

157 + 163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211


The representations of as the sum of one or more consecutive primes are shown in the following table.

Representations as the sum of one or more consecutive primes
  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 * 16 + {} {} {2} {3} {} {2+3, 5} {} {7} {3+5} {} {2+3+5} {11} {5+7} {13} {} {3+5+7}
1 * 16 + {} {2+3+5+7, 17} {7+11} {19} {} {} {} {5+7+11, 23} {11+13} {} {3+5+7+11} {} {2+3+5+7+11} {29} {13+17} {7+11+13, 31}
2 * 16 + {} {} {} {} {5+7+11+13, 17+19} {37} {} {3+5+7+11+13} {} {2+3+5+7+11+13, 11+13+17, 41} {19+23} {43} {} {} {} {47}
3 * 16 + #{?} = 1 #{?} = 1 {} {} #{?} = 1 {5 + 7 + 11 + 13 + 17, 53} {} {} #{?} = 1 {} #{?} = 1 #{?} = 2 #{?} = 2 #{?} = 1 {} {}
4 * 16 + {} {} {} #{?} = 2 #{?} = 1 {} {} #{?} = 2 #{?} = 2 #{?} = 1 {} #{?} = 1 {} #{?} = 1 #{?} = 1 #{?} = 1
5 * 16 + {} {} {} #{?} = 3 #{?} = 1 {} {} {} #{?} = 1 #{?} = 1 #{?} = 2 {} {} {} {} #{?} = 1
6 * 16 + {} #{?} = 2 #{?} = 1 {0} #{?} = 2 #{?} = 2 #{?} = 1 #{?} = 1 {} {} {} #{?} = 1 {} #{?} = 2 {} {}
7 * 16 + {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?}
8 * 16 + {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?}
9 * 16 + {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?}
10 * 16 + {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?}
11 * 16 + {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?}
12 * 16 + {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?} {?}
13 * 16 + {} {} #{?} = 2 #{?} = 2 {} {} {} {} #{?} = 1 {} {} {} #{?} = 1 #{?} = 2 #{?} = 1 #{?} = 3
14 * 16 + {} {} {} #{?} = 1 #{?} = 2 #{?} = 1 {} {} {} #{?} = 2 {} #{?} = 1 #{?} = 1 {0} #{?} = 1 #{?} = 1
15 * 16 + #{?} = 3 #{?} = 1 {} #{?} = 1 {} {} {} {} {} {} {} #{?} = 3 #{?} = 1 #{?} = 1 {} {}

A007504 Smallest integer that can be expressed as the sum of consecutive primes, i.e. sum of first primes.

{2, 5, 10, 17, 28, 41, 58, 77, 100, 129, 160, 197, 238, 281, 328, 381, 440, 501, 568, 639, 712, 791, 874, 963, 1060, 1161, 1264, 1371, 1480, 1593, 1720, 1851, 1988, 2127, 2276, 2427, 2584, 2747, 2914, ...}

A054859 Smallest integer that can be expressed as the sum of consecutive primes in exactly ways.

{1, 2, 5, 41, 1151, 311, 34421, 218918, 3634531, 48205429, ...}

A060433 Smallest odd number that can be represented as the sum of one or more consecutive primes in ways. (a(0) should be 1, unless we change the definition to "smallest nonunit odd number" — Daniel Forgues 21:22, 3 November 2011 (UTC))

{9, 3, 5, 41, 1151, 311, 34421, 442019, 3634531, 48205429, ...}

A?????? Count of numbers which are the sum of one or more consecutive primes.

{5, 53, 480, 4809, 48538, 481999, 4794890, 47790460, ...}

A074192 Count of numbers which are not the sum of one or more consecutive primes.

{5, 47, 520, 5191, 51462, 518001, 5205110, 52209540, ...}

Ratio of count of numbers which are not the sum of one or more consecutive primes over count of numbers which are the sum of one or more consecutive primes.

{1, 0.88679245283018, 1.08333333333333, 1.07943439384487, 1.06024146029915, 1.07469310102303, 1.08555357891422, 1.09246782726092, ...}

Number of representations as the sum of one or more consecutive primes

The arithmetic mean of the number of representations of as the sum of one or more consecutive primes is asymptotic to , i.e.

which means that the geometric mean of converges to 2, i.e.

A054845 Number of ways of representing as the sum of one or more consecutive primes.

{0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 0, 0, 2, 1, 0, 1, 0, 1, 1, 1, 2, 0, 0, 0, 0, 2, 1, 0, 1, 0, 3, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 2, 0, 0, 1, 0, 1, 2, 2, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 2, 2, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 3, 1, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 1, 0, 2, 1, 0, 2, 2, ...}

Representations as the sum of two or more consecutive primes

A084143 Number of partitions of into a sum of two or more consecutive primes.

{0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 1, ...}

Representations of primes as the sum of two or more consecutive primes

For example, 2011 (a prime number) is the sum of eleven consecutive primes ending in 211

2011 = 157 + 163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211

A067377 Primes expressible as the sum of (at least two) consecutive primes in at least 1 way.

{5, 17, 23, 31, 41, 53, 59, 67, 71, 83, 97, 101, 109, 127, 131, 139, 173, 181, 197, 199, 211, 223, 233, 251, 263, 269, 271, 281, 311, 331, 349, 353, 373, 379, 401, 421, 431, 439, 443, 449, 457, 463, 479, 487, 491, 499, 503, 523, 563, 587, 593, 607, 617, 631, ...}

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