2 = 1/2 + 2/4 + 3/8 + 4/16 + 5/32…

sum(n^p / 2ⁿ)

2 = prime = sum(n / 2ⁿ)
6 = 2·3 = sum(n² / 2ⁿ)
26 = 2·13 = sum(n³ / 2ⁿ)
150 = 2·3·5² = sum(n⁴ / 2ⁿ)
1082 = 2·541 = sum(n⁵ / 2ⁿ)
9366 = 2·3·7·223 = sum(n⁶ / 2ⁿ)
94586 = 2·47293 = sum(n⁷ / 2ⁿ)
1091670 = 2·3·5·36389 = sum(n⁸ / 2ⁿ)
14174522 = 2·7087261 = sum(n⁹ / 2ⁿ)
204495126 = 2·3·11·41·75571 = sum(n¹⁰ / 2ⁿ)

• A000629 Number of necklaces of partitions of n+1 labeled beads.

1, 2, 6, 26, 150, 1082, 9366, 94586, 1091670, 14174522, 204495126, 3245265146, 56183135190, 1053716696762, 21282685940886, 460566381955706, 10631309363962710, 260741534058271802, 6771069326513690646, 185603174638656822266, 5355375592488768406230

• moniliform ← French moniliforme (1800 or earlier) ← classical Latin monīle necklace

sum(n^p / 3ⁿ)

3/4 = prime / (2²) = sum(n / 3ⁿ)
3/2 = prime / prime = sum(n² / 3ⁿ)
33/8 = (3·11) / (2³) = sum(n³ / 3ⁿ)
15 = 3·5 = sum(n⁴ / 3ⁿ)
273/4 = (3·7·13) / (2²) = sum(n⁵ / 3ⁿ)
1491/4 = (3·7·71) / (2²) = sum(n⁶ / 3ⁿ)
38001/16 = (3·53·239) / (2⁴) = sum(n⁷ / 3ⁿ)
17295 = 3·5·1153 = sum(n⁸ / 3ⁿ)
566733/4 = (3·188911) / (2²) = sum(n⁹ / 3ⁿ)
2579313/2 = (3·11·47·1663) / prime = sum(n¹⁰ / 3ⁿ)

sum(n^p / 4ⁿ)

4/9 = (2²) / (3²) = sum(n / 4ⁿ)
20/27 = (2²·5) / (3³) = sum(n² / 4ⁿ)
44/27 = (2²·11) / (3³) = sum(n³ / 4ⁿ)
380/81 = (2²·5·19) / (3⁴) = sum(n⁴ / 4ⁿ)
4108/243 = (2²·13·79) / (3⁵) = sum(n⁵ / 4ⁿ)
17780/243 = (2²·5·7·127) / (3⁵) = sum(n⁶ / 4ⁿ)
269348/729 = (2²·17²·233) / (3⁶) = sum(n⁷ / 4ⁿ)
4663060/2187 = (2²·5·107·2179) / (3⁷) = sum(n⁸ / 4ⁿ)
10091044/729 = (2²·2522761) / (3⁶) = sum(n⁹ / 4ⁿ)
218374420/2187 = (2²·5·11·23·103·419) / (3⁷) = sum(n¹⁰ / 4ⁿ)

sum(n^p / 5ⁿ)

5/16 = prime / (2⁴) = sum(n / 5ⁿ)
15/32 = (3·5) / (2⁵) = sum(n² / 5ⁿ)
115/128 = (5·23) / (2⁷) = sum(n³ / 5ⁿ)
285/128 = (3·5·19) / (2⁷) = sum(n⁴ / 5ⁿ)
3535/512 = (5·7·101) / (2⁹) = sum(n⁵ / 5ⁿ)
26355/1024 = (3·5·7·251) / (2¹⁰) = sum(n⁶ / 5ⁿ)
458555/4096 = (5·91711) / (2¹²) = sum(n⁷ / 5ⁿ)
1139685/2048 = (3·5·75979) / (2¹¹) = sum(n⁸ / 5ⁿ)
25492435/8192 = (5·17·443·677) / (2¹³) = sum(n⁹ / 5ⁿ)
316786305/16384 = (3·5·11·1919917) / (2¹⁴) = sum(n¹⁰ / 5ⁿ)

sum(n^p / 6ⁿ)

6/25 = (2·3) / (5²) = sum(n / 6ⁿ)
42/125 = (2·3·7) / (5³) = sum(n² / 6ⁿ)
366/625 = (2·3·61) / (5⁴) = sum(n³ / 6ⁿ)
4074/3125 = (2·3·7·97) / (5⁵) = sum(n⁴ / 6ⁿ)
11334/3125 = (2·3·1889) / (5⁵) = sum(n⁵ / 6ⁿ)
189714/15625 = (2·3·7·4517) / (5⁶) = sum(n⁶ / 6ⁿ)
3706518/78125 = (2·3·181·3413) / (5⁷) = sum(n⁷ / 6ⁿ)
82749954/390625 = (2·3·7·1970237) / (5⁸) = sum(n⁸ / 6ⁿ)
2078250726/1953125 = (2·3·31·1061·10531) / (5⁹) = sum(n⁹ / 6ⁿ)
11598884682/1953125 = (2·3·7·11·23²·47459) / (5⁹) = sum(n¹⁰ / 6ⁿ)

sum(n^p / 7ⁿ)

7/36 = prime / (2²·3²) = sum(n / 7ⁿ)
7/27 = prime / (3³) = sum(n² / 7ⁿ)
91/216 = (7·13) / (2³·3³) = sum(n³ / 7ⁿ)
70/81 = (2·5·7) / (3⁴) = sum(n⁴ / 7ⁿ)
2149/972 = (7·307) / (2²·3⁵) = sum(n⁵ / 7ⁿ)
3311/486 = (7·11·43) / (2·3⁵) = sum(n⁶ / 7ⁿ)
285929/11664 = (7·40847) / (2⁴·3⁶) = sum(n⁷ / 7ⁿ)
220430/2187 = (2·5·7·47·67) / (3⁷) = sum(n⁸ / 7ⁿ)
1359337/2916 = (7·17·11423) / (2²·3⁶) = sum(n⁹ / 7ⁿ)
5239157/2187 = (7·11·68041) / (3⁷) = sum(n¹⁰ / 7ⁿ)

sum(n^p / 8ⁿ)

8/49 = (2³) / (7²) = sum(n / 8ⁿ)
72/343 = (2³·3²) / (7³) = sum(n² / 8ⁿ)
776/2401 = (2³·97) / (7⁴) = sum(n³ / 8ⁿ)
10440/16807 = (2³·3²·5·29) / (7⁵) = sum(n⁴ / 8ⁿ)
174728/117649 = (2³·21841) / (7⁶) = sum(n⁵ / 8ⁿ)
3525192/823543 = (2³·3²·11·4451) / (7⁷) = sum(n⁶ / 8ⁿ)
11870648/823543 = (2³·41·36191) / (7⁷) = sum(n⁷ / 8ⁿ)
319735800/5764801 = (2³·3²·5²·19·9349) / (7⁸) = sum(n⁸ / 8ⁿ)
9686934584/40353607 = (2³·1210866823) / (7⁹) = sum(n⁹ / 8ⁿ)
326084753016/282475249 = (2³·3²·11·16273·25301) / (7¹⁰) = sum(n¹⁰ / 8ⁿ)

sum(n^p / 9ⁿ)

9/64 = (3²) / (2⁶) = sum(n / 9ⁿ)
45/256 = (3²·5) / (2⁸) = sum(n² / 9ⁿ)
531/2048 = (3²·59) / (2¹¹) = sum(n³ / 9ⁿ)
1935/4096 = (3²·5·43) / (2¹²) = sum(n⁴ / 9ⁿ)
34983/32768 = (3²·13²·23) / (2¹⁵) = sum(n⁵ / 9ⁿ)
381465/131072 = (3²·5·7²·173) / (2¹⁷) = sum(n⁶ / 9ⁿ)
9725787/1048576 = (3²·67·127²) / (2²⁰) = sum(n⁷ / 9ⁿ)
35420535/1048576 = (3²·5·787123) / (2²⁰) = sum(n⁸ / 9ⁿ)
1160703963/8388608 = (3²·47·409·6709) / (2²³) = sum(n⁹ / 9ⁿ)
21129845715/33554432 = (3²·5·11·2213·19289) / (2²⁵) = sum(n¹⁰ / 9ⁿ)

sum(n^p / 10ⁿ)

10/81 = (2·5) / (3⁴) = sum(n / 10ⁿ)
110/729 = (2·5·11) / (3⁶) = sum(n² / 10ⁿ)
470/2187 = (2·5·47) / (3⁷) = sum(n³ / 10ⁿ)
7370/19683 = (2·5·11·67) / (3⁹) = sum(n⁴ / 10ⁿ)
142870/177147 = (2·5·7·13·157) / (3¹¹) = sum(n⁵ / 10ⁿ)
1114190/531441 = (2·5·7·11·1447) / (3¹²) = sum(n⁶ / 10ⁿ)
30495890/4782969 = (2·5·3049589) / (3¹⁴) = sum(n⁷ / 10ⁿ)
953934190/43046721 = (2·5·11·569·15241) / (3¹⁶) = sum(n⁸ / 10ⁿ)
3728765410/43046721 = (2·5·372876541) / (3¹⁶) = sum(n⁹ / 10ⁿ)
145739620510/387420489 = (2·5·11·1324905641) / (3¹⁸) = sum(n¹⁰ / 10ⁿ)