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The Cullen numbers up to 10

1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, 229377, 491521, 1048577, 2228225, 4718593, 9961473, 20971521, 44040193, 92274689, 192937985, 402653185, 838860801, 1744830465, 3623878657, 7516192769, 15569256449, 32212254721, 66571993089, 137438953473, 283467841537, 584115552257, 1202590842881, 2473901162497, 5085241278465, 10445360463873, 21440476741633, 43980465111041, 90159953477633, 184717953466369, 378231999954945, 774056185954305.

__Distribution of the remainders__ when the numbers in this family are divided by *n*=2, 3,..., 11. (I took into account 100000 values, from 1 to 5⋅10^{30107}).

n\r | 0 | 1 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

2 | 0 | 100000 | 2 | ||||||||

3 | 33334 | 33334 | 33332 | 3 | |||||||

4 | 0 | 99999 | 0 | 1 | 4 | ||||||

5 | 20000 | 20000 | 20000 | 20000 | 20000 | 5 | |||||

6 | 0 | 33334 | 0 | 33334 | 0 | 33332 | 6 | ||||

7 | 14286 | 14286 | 14286 | 14286 | 14284 | 14286 | 14286 | 7 | |||

8 | 0 | 99999 | 0 | 1 | 0 | 0 | 0 | 0 | 8 | ||

9 | 11111 | 11112 | 11111 | 11111 | 11110 | 11110 | 11112 | 11112 | 11111 | 9 | |

10 | 0 | 20000 | 0 | 20000 | 0 | 20000 | 0 | 20000 | 0 | 20000 | 10 |

11 | 9092 | 9091 | 9090 | 9093 | 9090 | 9090 | 9091 | 9091 | 9090 | 9091 | 9091 |

Imagine to divide the members of this family by a number *n* and compute the remainders. Should they be uniformly distributed, each remainder from 0 to *n*-1 would be obtained in about (1/*n*)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/*n*.

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