A product of primes in a particular arithmetic-geometric progression. more
The first 600 Zeisel numbers :
105,
1419,
1729,
1885,
4505,
5719,
15387,
24211,
25085,
27559,
31929,
54205,
59081,
114985,
207177,
208681,
233569,
287979,
294409,
336611,
353977,
448585,
507579,
982513,
1012121,
1073305,
1242709,
1485609,
2089257,
2263811,
2953711,
3077705,
3506371,
3655861,
3812599,
3973085,
4293793,
4648261,
4813879,
5069629,
6173179,
6253085,
6622609,
6985249,
7355239,
7355671,
7558219,
8011459,
8413179,
8444431,
8712985,
8835799,
9271805,
9773731,
10235499,
13534129,
14662681,
15411785,
16596371,
18175361,
18578113,
18651417,
19827641,
20138379,
20771801,
22549639,
23691481,
25711399,
26000605,
26758057,
31852931,
32812089,
32894743,
34179391,
34878529,
35347159,
37605385,
38596273,
41305159,
42501439,
43055057,
44544219,
46999705,
48476449,
49982899,
52691801,
53399449,
54177877,
55902529,
56052361,
59267089,
65237577,
68633979,
69207769,
71550913,
71657209,
72457085,
72730439,
76724569,
78225589,
79796761,
84280585,
87393129,
92458529,
92835667,
96916279,
96931639,
100085785,
104966471,
118901521,
126893905,
127090931,
130692329,
133800661,
134574729,
161164441,
162913085,
167195509,
170383081,
172947529,
176116299,
177055201,
185245273,
195850411,
197916193,
199708657,
210509529,
212122639,
213701405,
216821881,
221042353,
222931549,
228842209,
237646105,
240479779,
249925249,
257519143,
265265329,
265786107,
266032009,
268578181,
269545249,
272174369,
290770505,
291335089,
306333559,
320237979,
321767447,
323476777,
332898769,
342116741,
367224619,
372526121,
375615073,
378606709,
392605449,
418637557,
426689979,
428108921,
443880589,
458576371,
483824809,
513769209,
559874269,
572905085,
578178931,
599938361,
604611019,
607692079,
627977681,
644074039,
656960585,
662998921,
667235531,
668997041,
671405827,
732641959,
758056531,
763651289,
770382685,
781296905,
790296729,
820192231,
826048329,
856970353,
872159101,
873571459,
939947009,
953055097,
959646507,
970578289,
976228129,
994732211,
995762585,
1006956091,
1007749849,
1017436249,
1020225061,
1023780151,
1056066169,
1075122185,
1102226791,
1129119819,
1136972771,
1140806899,
1151271109,
1158590105,
1164151711,
1191610211,
1244556249,
1262588371,
1268239597,
1276450585,
1299963601,
1306276309,
1324715137,
1348964401,
1412889731,
1414608769,
1449600967,
1455928891,
1495619759,
1518477085,
1539866809,
1586031409,
1614365473,
1615067791,
1637505339,
1640582839,
1640591017,
1641183305,
1659996371,
1667082529,
1674819721,
1675383631,
1715605681,
1743564961,
1743855049,
1748351011,
1793680361,
1813376689,
1826139169,
1886676121,
2016369161,
2038542729,
2050620661,
2154988109,
2190901987,
2194078033,
2219965885,
2225685529,
2242036505,
2279335659,
2279884349,
2289981241,
2301745249,
2307778289,
2309125085,
2400112121,
2423890585,
2659890241,
2709852103,
2748840281,
2812661659,
2960089189,
2972190761,
2974949209,
3020799259,
3164104819,
3221257085,
3394026439,
3472944211,
3481730585,
3492435179,
3543047209,
3559229569,
3632325085,
3728382589,
3764839331,
3871130937,
3916728211,
4110034585,
4138778957,
4143291485,
4147129841,
4170899461,
4183998729,
4263791959,
4277379709,
4441014181,
4547166601,
4569785479,
4859411289,
4978601097,
5038919979,
5165487199,
5180590981,
5213778037,
5226353885,
5282265793,
5339894761,
5339947879,
5542870939,
5568193961,
5666384905,
5694879799,
5789613817,
5851234681,
5879848969,
6044255485,
6087249305,
6101471689,
6117620389,
6154662709,
6192117463,
6217453085,
6234117391,
6278206481,
6349784379,
6501404989,
6517127953,
6585826301,
6650251489,
6708383011,
6760613399,
6914878129,
6930157429,
7010842411,
7044296281,
7130443159,
7200759409,
7330282961,
7348575649,
7469664729,
7509370121,
7534916761,
7559985223,
7730810221,
7778780353,
7865839129,
7871298611,
8187154585,
8243864569,
8277624793,
8348185819,
8641410913,
8657319259,
8691633217,
8926716889,
8952813689,
9013446721,
9054349841,
9067727787,
9110311489,
9115777783,
9153216689,
9225913085,
9278421769,
9478765447,
9531357049,
9624742921,
9652464361,
9692605801,
9810113209,
9855654089,
9856200761,
9925219805,
10077765769,
10167084209,
10184909659,
10283096409,
10311833251,
10479201657,
10611175909,
10698325639,
10735275881,
10821351979,
10838219401,
11160127441,
11247820759,
11346205609,
11366933131,
11585424313,
11732139049,
11746422361,
11759151385,
11986605961,
12098215429,
12171272059,
12222787609,
12400061059,
12440346185,
12479816371,
12509980499,
12564570721,
12690860379,
12735900289,
12770699239,
12986373961,
13029841441,
13078092481,
13079177569,
13254648793,
13259685129,
13510685885,
13658595497,
13725848369,
13955446205,
14040243481,
14084231019,
14221248371,
14447821929,
14480424841,
14629980199,
14684898709,
14942269977,
15030947485,
15031011241,
15452754269,
15498062839,
15518134369,
15642681619,
15811322689,
15920790281,
16113640667,
16122019369,
16222456009,
16227462057,
16292245177,
16326149405,
16399006681,
16501875337,
16527263369,
16896808347,
16974900769,
17086366985,
17130058219,
17141908699,
17186116177,
17460116023,
17832756943,
18511865209,
18859456585,
18870656953,
18951192605,
18955516609,
18999242011,
19443808993,
19487842291,
19715531561,
19742778857,
19898997449,
20042866351,
20129056711,
20522943379,
20987583819,
21062533889,
21515221081,
21820353731,
22024971809,
22266935979,
22727176201,
22765299785,
22785447691,
23228252929,
23382858919,
23426237069,
23691113491,
23926664605,
23966318489,
24069892121,
24117279229,
24281829339,
24358575485,
24441518809,
24629028747,
24644476423,
25349312585,
25455578113,
25619904919,
25773345613,
26168577769,
26933878399,
26966811129,
27041372471,
27059617085,
27278026129,
27320146681,
27339094617,
27496251451,
27736355089,
27995264983,
28010687569,
28166983489,
28361573785,
28398933289,
28608920089,
28877917873,
29115769789,
29191946393,
29329954081,
29481412559,
29611230121,
29842766619,
29961353569,
30328881305,
30457991929,
30513088039,
30961811005,
31252080777,
31297807729,
32028855251,
32123162185,
32916256809,
32982255649,
33373559569,
33382122409,
33676321771,
33687886009,
33715496519,
34325216809,
34387241729,
34633253089,
34945134205,
35176894513,
35240819101,
35361280585,
35376563089,
35732398969,
35744803179,
35894813221,
36062829937,
36314127739,
36622579961,
36623071489,
36772274585,
36811632961,
36825331879,
36941867299,
36949468249,
37001838329,
37035472369,
37049946529,
37329650461,
37701550321,
38177322479,
38356861411,
38368399609,
38427198769,
38930058649,
38967269307,
40124858209,
40284176359,
40334231071,
40405589979,
40406711489,
40509227011,
40598826217,
40892779419,
40928444929,
41475289633,
41882878241,
41999815513,
42004845697,
42005625085,
42091988743,
42321696529,
42718639177,
42740642131,
42749039089,
43015831159,
43108560905,
43197819211,
44319075631,
44441688649,
44480201239,
45135348889,
45275069911,
45426093931,
45445771481,
45892640089,
47523935561,
47703457099,
48339438679,
48674377081,
49228857627,
49608449885,
49863902599,
50063838729,
50379089719,
50873669009,
50896218997,
50909186161,
51195800585,
51908793949,
52475323159,
53544900199,
54285827005,
54843712585,
55583176121.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 9607 values, from 105 to 999965521635409).
n\r | 0 | 1 |
2 | 0 | 9607 | 2 |
3 | 615 | 7020 | 1972 | 3 |
4 | 0 | 6830 | 0 | 2777 | 4 |
5 | 765 | 3103 | 824 | 633 | 4282 | 5 |
6 | 0 | 7020 | 0 | 615 | 0 | 1972 | 6 |
7 | 759 | 3658 | 1193 | 248 | 1495 | 1099 | 1155 | 7 |
8 | 0 | 5490 | 0 | 1504 | 0 | 1340 | 0 | 1273 | 8 |
9 | 0 | 4420 | 647 | 0 | 1419 | 697 | 615 | 1181 | 628 | 9 |
10 | 0 | 3103 | 0 | 633 | 0 | 765 | 0 | 824 | 0 | 4282 | 10 |
11 | 581 | 1391 | 987 | 786 | 1194 | 1042 | 752 | 440 | 769 | 879 | 786 |
A pictorial representation of the table above
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.