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# Representations as the sum of one or more consecutive primes

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 ${\displaystyle \scriptstyle n,\,n\,\geq \,0,\,}$ 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 ${\displaystyle \scriptstyle n,\,n\,\geq \,1,\,}$ consecutive primes, i.e. sum of first ${\displaystyle \scriptstyle n,\,n\,\geq \,1,\,}$ 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 ${\displaystyle \scriptstyle n,\,n\,\geq \,0,\,}$ 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 ${\displaystyle \scriptstyle n,\,n\,\geq \,0,\,}$ 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 ${\displaystyle \scriptstyle 0\,\leq \,m\,<\,10^{n},\,n\,\geq \,1,\,}$ which are the sum of one or more consecutive primes.

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

A074192 Count of numbers ${\displaystyle \scriptstyle 0\,\leq \,m\,<\,10^{n},\,n\,\geq \,1,\,}$ which are not the sum of one or more consecutive primes.

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

Ratio of count of numbers ${\displaystyle \scriptstyle 0\,\leq \,m\,<\,10^{n},\,n\,\geq \,1,\,}$ which are not the sum of one or more consecutive primes over count of numbers ${\displaystyle \scriptstyle 0\,\leq \,m\,<\,10^{n},\,n\,\geq \,1,\,}$ 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 ${\displaystyle \scriptstyle g(n)\,}$ of representations of ${\displaystyle \scriptstyle n\,}$ as the sum of one or more consecutive primes is asymptotic to ${\displaystyle \scriptstyle \log 2\,}$, i.e.

${\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{k=1}^{n}g(k)=\log 2,\,}$

which means that the geometric mean of ${\displaystyle \scriptstyle e^{g(k)}\,}$ converges to 2, i.e.

${\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{\prod _{k=1}^{n}e^{g(k)}}}=2.\,}$

A054845 Number of ways of representing ${\displaystyle \scriptstyle n,\,n\,\geq \,0,\,}$ 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 ${\displaystyle \scriptstyle n,\,n\,\geq \,1,\,}$ 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, ...}