W

 Will Nicholes

List of prime signatures

Below is a list of the first 400 prime signatures.

The OEIS column shows the Online Encyclopedia of Integer Sequences entry (if any) for the integers belonging to the specified prime signature. For example, row #8’s prime signature is {1,1,1}, so it links to OEIS sequence A007304, which lists integers with the prime signature of {1,1,1}.

The smaller entries with an asterisk (*) indicate an inexact match in OEIS: while the first hundred or so terms shown there match the prime signature listed, later terms include integers with a different prime signature.

The column labeled First displays the smallest integer with that row’s prime signature (OEIS A025487).

Similarly, the column labeled Second displays the second smallest integer with that row’s prime signature (OEIS A077560).

(For additional integers with a given prime signature — third, fourth, etc. — see the prime signature table page.)

The Prime fact. column shows the prime factorization corresponding to the prime signature, where each letter (p, q, etc.) represents a distinct prime number.

The First odd column displays the first odd integer with that row’s prime signature. The numbers in this column are a re-ordering of the numbers in OEIS A147516.

The PD column lists the number of Prime Divisors (OEIS A036041.) For example, {1,2,3} has 6 prime divisors (1 + 2 + 3).

The DF column lists the number of Distinct Prime Factors (OEIS A061394.)

Lastly, the Diff column lists the difference between the smallest integer with this prime signature and the smallest integer with the previous row’s prime signature (OEIS A178799.)

The table is broken into 4 sections to better fit in the browser window.

 

Prime signature list, part 1

#OEISPrime signatureFirstSecondPrime fact.First oddP DD FDiff
0-{}1-p0100-
1A000040{1}23p 3111
2A001248{2}49p29212
3A006881{1,1}610pq 15222
4A030078{3}827p327312
5A054753{1,2}1218p2q45324
6A030514{4}1681p481414
7A065036{1,3}2440p3q135428
8A007304{1,1,1}3042pqr 105336
9A050997{5}32243p5243512
10A085986{2,2}36100p2q2225424
11A178739{1,4}4880p4q4055212
12A085987{1,1,2}6084p2qr3154312
13A030516{6}64729p6729614
14A143610{2,3}72108p3q2675528
15A178740{1,5}96160p5q12156224
16-{1,1,3}120168p3qr9455324
17A092759{7}1282187p72187718
18A030633*{2,4}144324p4q220256216
19-{1,2,2}180252p2q2r15755336
20A030632*{1,6}192320p6q36457212
21A046386{1,1,1,1}210330pqrs 11554418
22A162142{3,3}2161000p3q33375626
23-{1,1,4}240336p4qr28356324
24-{8}2566561p865618116
25-{2,5}288800p5q260757232
26A163569{1,2,3}360504p3q2r47256372
27-{1,7}384640p7q109358224
28-{1,1,1,2}420630p2qrs34655436
29-{3,4}432648p4q3101257212
30-{1,1,5}480672p5qr85057348
31-{9}51219683p9196839132
32A137484*{2,6}5761600p6q2182258264
33-{1,2,4}7201008p4q2r1417573144
34-{1,8}7681280p8q328059248
35-{1,1,1,3}8401320p3qrs103956472
36-{3,5}8641944p5q3303758224
37A162143{2,2,2}9001764p2q2r2110256336
38-{1,1,6}9601344p6qr255158360
39A030629{10}102459049p105904910164
40-{1,3,3}10801512p3q3r236257356
41-{2,7}11523200p7q2546759272
42-{1,1,2,2}12601980p2q2rs1732564108
43A137488*{4,4}129610000p4q4506258236
44-{1,2,5}14402016p5q2r4252583144
45-{1,9}15362560p9q9841510296
46-{1,1,1,4}16802640p4qrs3118574144
47-{3,6}17285832p6q3911259248
48-{2,2,3}18002700p3q2r2330757372
49-{1,1,7}19202688p7qr7654593120
50A079395{11}2048177147p11177147111128
51-{1,3,4}21603024p4q3r7087583112
52-{2,8}23046400p8q2164025102144
53A046387{1,1,1,1,1}23102730pqrst 15015556
54{1,1,2,3}25203780p3q2rs5197574210
55{4,5}25923888p5q41518759272
56{1,2,6}28804032p6q2r12757593288
57{1,10}30725120p10q295245112192
58{1,1,1,5}33605280p5qrs9355584288
59{3,7}345616000p7q327337510296

Prime signature list, part 2

#Prime signatureFirstSecondPrime fact.First oddP DD FDiff
60{2,2,4}36007056p4q2r29922583144
61{1,1,8}38405376p8qr229635103240
62{12}4096531441p12531441121256
63{1,3,5}43206048p5q3r21262593224
64{2,9}460812800p9q2492075112288
65{1,1,1,1,2}46205460p2qrst450456512
66{1,1,2,4}50407920p4q2rs15592584420
67{4,6}518411664p6q4455625102144
68{2,3,3}54009000p3q3r216537583216
69{1,2,7}57608064p7q2r382725103360
70{1,11}614410240p11q885735122384
71{1,2,2,2}63008820p2q2r2s12127574156
72{1,4,4}64809072p4q4r35437593180
73{1,1,1,6}672010560p6qrs28066594240
74{3,8}691232000p8q3820125112192
75{2,2,5}720014112p5q2r229767593288
76{1,1,3,3}756011880p3q3rs25987584360
77{1,1,9}768010752p9qr688905113120
78{5,5}7776100000p5q575937510296
79{13}81921594323p131594323131416
80{1,3,6}864012096p6q3r637875103448
81{2,10}921625600p10q21476225122576
82{1,1,1,1,3}924010920p3qrst1351357524
83{1,1,2,5}1008015840p5q2rs46777594840
84{4,7}1036834992p7q41366875112288
85{2,3,4}1080016200p4q3r249612593432
86{1,2,8}1152016128p8q2r1148175113720
87{1,12}1228820480p12q2657205132768
88{1,2,2,3}1260017640p3q2r2s36382584312
89{1,4,5}1296018144p5q4r1063125103360
90{1,1,1,7}1344021120p7qrs841995104480
91{3,9}1382464000p9q32460375122384
92{1,1,1,2,2}1386016380p2q2rst2252257536
93{2,2,6}1440028224p6q2r2893025103540
94{1,1,3,4}1512022680p4q3rs77962594720
95{1,1,10}1536021504p10qr2066715123240
96{5,6}1555223328p6q52278125112192
97{14}163844782969p144782969141832
98{1,3,7}1728024192p7q3r1913625113896
99{2,11}1843251200p11q244286751321152

Prime signature list, part 3

#Prime signatureFirstSecondPrime fact.P DD FDiff
100{1,1,1,1,4}1848021840p4qrst8548
101{1,1,2,6}2016031680p6q2rs1041680
102{4,8}20736104976p8q4122576
103{2,3,5}2160036000p5q3r2103864
104{1,2,9}2304032256p9q2r1231440
105{1,13}2457640960p13q1421536
106{1,2,2,4}2520035280p4q2r2s94624
107{1,4,6}2592036288p6q4r113720
108{1,1,1,8}2688042240p8qrs114960
109{3,3,3}2700074088p3q3r393120
110{3,10}27648128000p10q3132648
111{1,1,1,2,3}2772032760p3q2rst8572
112{2,2,7}2880056448p7q2r21131080
113{1,1,1,1,1,1}3003039270pqrstu 661230
114{1,1,3,5}3024047520p5q3rs104210
115{1,1,11}3072043008p11qr133480
116{5,7}3110469984p7q5122384
117{2,4,4}3240063504p4q4r21031296
118{15}3276814348907p15151368
119{1,3,8}3456048384p8q3r1231792
120{2,12}36864102400p12q21422304
121{1,1,1,1,5}3696043680p5qrst9596
122{1,2,3,3}3780052920p3q3r2s94840
123{1,5,5}3888054432p5q5r1131080
124{1,1,2,7}4032063360p7q2rs1141440
125{4,9}41472314928p9q41321152
126{2,3,6}4320072000p6q3r21131728
127{2,2,2,2}44100108900p2q2r2s284900
128{1,1,4,4}4536071280p4q4rs1041260
129{1,2,10}4608064512p10q2r133720
130{6,6}466561000000p6q6122576
131{1,14}4915281920p14q1522496
132{1,2,2,5}5040070560p5q2r2s1041248
133{1,4,7}5184072576p7q4r1231440
134{1,1,1,9}5376084480p9qrs1241920
135{3,3,4}5400081000p4q3r3103240
136{3,11}55296256000p11q31421296
137{1,1,1,2,4}5544065520p4q2rst95144
138{2,2,8}57600112896p8q2r21232160
139{1,1,1,1,1,2}6006078540p2qrstu762460
140{1,1,3,6}6048095040p6q3rs114420
141{1,1,12}6144086016p12qr143960
142{5,8}62208209952p8q5132768
143{2,4,5}6480097200p5q4r21132592
144{16}6553643046721p16161736
145{1,3,9}6912096768p9q3r1333584
146{1,1,2,2,2}6930081900p2q2r2st85180
147{2,13}73728204800p13q21524428
148{1,1,1,1,6}7392087360p6qrst105192
149{1,2,3,4}75600105840p4q3r2s1041680
150{1,5,6}77760108864p6q5r1232160
151{1,1,2,8}80640126720p8q2rs1242880
152{4,10}82944640000p10q41422304
153{1,1,1,3,3}8316098280p3q3rst95216
154{2,3,7}86400144000p7q3r21233240
155{2,2,2,3}88200132300p3q2r2s2941800
156{1,1,4,5}90720136080p5q4rs1142520
157{1,2,11}92160129024p11q2r1431440
158{6,7}93312139968p7q61321152
159{1,15}98304163840p15q1624992
160{1,2,2,6}100800141120p6q2r2s1142496
161{1,4,8}103680145152p8q4r1332880
162{1,1,1,10}107520168960p10qrs1343840
163{3,3,5}108000243000p5q3r3113480
164{3,12}110592512000p12q31522592
165{1,1,1,2,5}110880131040p5q2rst105288
166{2,2,9}115200225792p9q2r21334320
167{1,1,1,1,1,3}120120157080p3qrstu864920
168{1,1,3,7}120960190080p7q3rs124840
169{1,1,13}122880172032p13qr1531920
170{5,9}124416629856p9q51421536
171{2,4,6}129600254016p6q4r21235184
172{17}131072129140163p171711472
173{1,3,10}138240193536p10q3r1437168
174{1,1,2,2,3}138600163800p3q2r2st95360
175{2,14}147456409600p14q21628856
176{1,1,1,1,7}147840174720p7qrst115384
177{1,2,3,5}151200211680p5q3r2s1143360
178{1,5,7}155520217728p7q5r1334320
179{1,1,2,9}161280253440p9q2rs1345760
180{3,4,4}162000270000p4q4r3113720
181{4,11}1658881280000p11q41523888
182{1,1,1,3,4}166320196560p4q3rst105432
183{2,3,8}172800288000p8q3r21336480
184{2,2,2,4}176400396900p4q2r2s21043600
185{1,1,1,1,2,2}180180235620p2q2rstu863780
186{1,1,4,6}181440285120p6q4rs1241260
187{1,2,12}184320258048p12q2r1532880
188{6,8}186624419904p8q61422304
189{1,3,3,3}189000297000p3q3r3s1042376
190{2,5,5}194400381024p5q5r21235400
191{1,16}196608327680p16q1722208
192{1,2,2,7}201600282240p7q2r2s1244992
193{1,4,9}207360290304p9q4r1435760
194{1,1,1,11}215040337920p11qrs1447680
195{3,3,6}216000592704p6q3r3123960
196{3,13}2211841024000p13q31625184
197{1,1,1,2,6}221760262080p6q2rst115576
198{1,2,4,4}226800317520p4q4r2s1145040
199{2,2,10}230400451584p10q2r21433600
200{1,6,6}233280326592p6q6r1332880
201{1,1,1,1,1,4}240240314160p4qrstu966960
202{1,1,3,8}241920380160p8q3rs1341680
203{1,1,14}245760344064p14qr1633840
204{5,10}2488321889568p10q51523072
205{2,4,7}259200508032p7q4r213310368
206{18}262144387420489p181812944
207{2,2,3,3}264600441000p3q3r2s21042456
208{1,1,5,5}272160427680p5q5rs1247560
209{1,3,11}276480387072p11q3r1534320
210{1,1,2,2,4}277200327600p4q2r2st105720
211{7,7}27993610000000p7q71422736
212{2,15}294912819200p15q217214976
213{1,1,1,1,8}295680349440p8qrst125768
214{1,2,3,6}302400423360p6q3r2s1246720
215{1,5,8}311040435456p8q5r1438640
216{1,1,2,10}322560506880p10q2rs14411520
217{3,4,5}324000486000p5q4r31231440
218{4,12}3317762560000p12q41627776
219{1,1,1,3,5}332640393120p5q3rst115864
220{2,3,9}345600576000p9q3r214312960
221{2,2,2,5}352800871200p5q2r2s21147200
222{1,1,1,1,2,3}360360471240p3q2rstu967560
223{1,1,4,7}362880570240p7q4rs1342520
224{1,2,13}368640516096p13q2r1635760
225{6,9}3732481259712p9q61524608
226{1,3,3,4}378000567000p4q3r3s1144752
227{2,5,6}388800583200p6q5r213310800
228{1,17}393216655360p17q1824416
229{1,2,2,8}403200564480p8q2r2s1349984
230{1,4,10}414720580608p10q4r15311520
231{1,1,2,3,3}415800491400p3q3r2st1051080
232{1,1,1,12}430080675840p12qrs15414280
233{3,3,7}4320001185408p7q3r31331920
234{3,14}4423682048000p14q317210368
235{1,1,1,2,7}443520524160p7q2rst1251152
236{1,2,4,5}453600635040p5q4r2s12410080
237{2,2,11}460800903168p11q2r21537200
238{1,6,7}466560653184p7q6r1435760
239{1,1,1,1,1,5}480480628320p5qrstu10613920
240{1,1,3,9}483840760320p9q3rs1443360
241{1,2,2,2,2}485100573300p2q2r2s2t951260
242{1,1,15}491520688128p15qr1736420
243{5,11}4976645668704p11q51626144
244{1,1,1,4,4}498960589680p4q4rst1151296
245{1,1,1,1,1,1,1}510510570570pqrstuv 7711550
246{2,4,8}5184001016064p8q4r21437890
247{19}5242881162261467p191915888
248{2,2,3,4}529200793800p4q3r2s21144912
249{1,1,5,6}544320816480p6q5rs13415120
250{1,3,12}552960774144p12q3r1638640
251{1,1,2,2,5}554400655200p5q2r2st1151440
252{7,8}559872839808p8q71525472
253{2,16}5898241638400p16q218229952
254{1,1,1,1,9}591360698880p9qrst1351536
255{1,2,3,7}604800846720p7q3r2s13413440
256{1,5,9}622080870912p9q5r15317280
257{1,1,2,11}6451201013760p11q2rs15423040
258{3,4,6}6480001080000p6q4r31332880
259{4,13}6635525120000p13q417215552
260{1,1,1,3,6}665280786240p6q3rst1251728
261{2,3,10}6912001152000p10q3r215325920
262{2,2,2,6}7056001742400p6q2r2s212414400
263{1,1,1,1,2,4}720720942480p4q2rstu10615120
264{1,1,4,8}7257601140480p8q4rs1445040
265{1,2,14}7372801032192p14q2r17311520
266{6,10}7464963779136p10q61629216
267{1,3,3,5}7560001188000p5q3r3s1249504
268{2,5,7}7776001524096p7q5r214321600
269{1,18}7864321310720p18q1928832
270{1,2,2,9}8064001128960p9q2r2s14419968
271{4,4,4}8100003111696p4q4r41233600
272{1,4,11}8294401161216p11q4r16319440
273{1,1,2,3,4}831600982800p4q3r2st1152160
274{1,1,1,13}8601601351680p13qrs16428560
275{3,3,8}8640002370816p8q3r31433840
276{3,15}8847364096000p15q318220736
277{1,1,1,2,8}8870401048320p8q2rst1352304
278{1,1,1,2,2,2}9009001178100p2q2r2stu9613860
279{1,2,4,6}9072001270080p6q4r2s1346300
280{2,2,12}9216001806336p12q2r216314400
281{1,6,8}9331201306368p8q6r15311520
282{1,1,1,1,1,6}9609601256640p6qrstu11627840
283{1,1,3,10}9676801520640p10q3rs1546720
284{1,2,2,2,3}9702001146600p3q2r2s2t1052520
285{3,5,5}9720002667168p5q5r31331800
286{1,1,16}9830401376256p16qr18311040
287{5,12}99532812800000p12q517212288
288{1,1,1,4,5}9979201179360p5q4rst1252592
289{1,1,1,1,1,1,2}10210201141140p2qrstuv8723100
290{2,4,9}10368002032128p9q4r215315780
291{20}10485763486784401p2020111776
292{2,2,3,5}10584001764000p5q3r2s21249824
293{1,1,1,1,3,3}10810801413720p3q3rstu10622680
294{1,1,5,7}10886401710720p7q5rs1447560
295{1,3,13}11059201548288p13q3r17317280
296{1,1,2,2,6}11088001310400p6q2r2st1252880
297{7,9}11197442519424p9q716210944
298{1,3,4,4}11340001782000p4q4r3s12414256
299{2,6,6}11664002286144p6q6r214332400
300{2,17}11796483276800p17q219213248
301{1,1,1,1,10}11827201397760p10qrst1453072
302{1,2,3,8}12096001693440p8q3r2s14426880
303{1,5,10}12441601741824p10q5r16334560
304{1,1,2,12}12902402027520p12q2rs16446080
305{3,4,7}12960002160000p7q4r31435760
306{2,3,3,3}13230001852200p3q3r3s211427000
307{4,14}132710410240000p14q41824104
308{1,1,1,3,7}13305601572480p7q3rst1353456
309{1,2,5,5}13608001905120p5q5r2s13430240
310{2,3,11}13824002304000p11q3r216321600
311{1,7,7}13996801959552p7q7r15317280
312{2,2,2,7}14112003484800p7q2r2s213411520
313{1,1,1,1,2,5}14414401884960p5q2rstu11630240
314{1,1,4,9}14515202280960p9q4rs15410080
315{1,2,15}14745602064384p15q2r18323040
316{6,11}149299211337408p11q617218432
317{1,3,3,6}15120002376000p6q3r3s13419008
318{2,5,8}15552003048192p8q5r215343200
319{1,19}15728642621440p19q20217664
320{2,2,4,4}15876003920400p4q4r2s212414736
321{1,2,2,10}16128002257920p10q2r2s15425200
322{4,4,5}16200002430000p5q4r41337200
323{1,1,6,6}16329602566080p6q6rs14412960
324{1,4,12}16588802322432p12q4r17325920
325{1,1,2,3,5}16632001965600p5q3r2st1254320
326{8,8}1679616100000000p8q816216416
327{1,1,1,14}17203202703360p14qrs17440704
328{3,3,9}17280004741632p9q3r31537680
329{3,16}17694728192000p16q319241472
330{1,1,1,2,9}17740802096640p9q2rst1454608
331{1,1,1,2,2,3}18018002356200p3q2r2stu10627720
332{1,2,4,7}18144002540160p7q4r2s14412600
333{2,2,13}18432003612672p13q2r217328800
334{1,6,9}18662402612736p9q6r16323040
335{1,1,1,1,1,7}19219202513280p7qrstu12655680
336{1,1,3,11}19353603041280p11q3rs16413440
337{1,2,2,2,4}19404002293200p4q2r2s2t1155040
338{3,5,6}19440002916000p6q5r31433600
339{1,1,17}19660802752512p17qr19322080
340{5,13}199065625600000p13q518224576
341{1,1,1,4,6}19958402358720p6q4rst1355184
342{1,1,1,1,1,1,3}20420402282280p3qrstuv9746200
343{2,4,10}20736004064256p10q4r216331560
344{1,1,3,3,3}20790002457000p3q3r3st1155400
345{21}209715210460353203p2121118152
346{2,2,3,6}21168003528000p6q3r2s213419648
347{1,1,1,1,3,4}21621602827440p4q3rstu11645360
348{1,1,5,8}21772803421440p8q5rs15415120
349{1,3,14}22118403096576p14q3r18334560
350{1,1,2,2,7}22176002620800p7q2r2st1355760
351{7,10}22394887558272p10q717221888
352{1,3,4,5}22680003402000p5q4r3s13428512
353{2,6,7}23328003499200p7q6r215364800
354{2,18}23592966553600p18q220226496
355{1,1,1,1,11}23654402795520p11qrst1556144
356{1,2,3,9}24192003386880p9q3r2s15453760
357{1,5,11}24883203483648p11q5r17369120
358{1,1,2,4,4}24948002948400p4q4r2st1256480
359{1,1,2,13}25804804055040p13q2rs17485680

Prime signature list, part 4

#Prime signatureFirstPrime fact.P DD FDiff
360{3,4,8}2592000p8q4r315311520
361{2,3,3,4}2646000p4q3r3s212454000
362{4,15}2654208p15q41928208
363{1,1,1,3,8}2661120p8q3rst1456912
364{1,2,5,6}2721600p6q5r2s14460480
365{2,3,12}2764800p12q3r217343200
366{1,7,8}2799360p8q7r16334560
367{2,2,2,8}2822400p8q2r2s214423040
368{1,1,1,1,2,6}2882880p6q2rstu12660480
369{1,1,4,10}2903040p10q4rs16420160
370{1,2,2,3,3}2910600p3q3r2s2t1157560
371{1,2,16}2949120p16q2r19338520
372{6,12}2985984p12q618236864
373{1,1,1,5,5}2993760p5q5rst1357776
374{1,3,3,7}3024000p7q3r3s14430240
375{1,1,1,1,1,2,2}3063060p2q2rstuv9739060
376{2,5,9}3110400p9q5r216347340
377{1,20}3145728p20q21235328
378{2,2,4,5}3175200p5q4r2s213429472
379{1,2,2,11}3225600p11q2r2s16450400
380{4,4,6}3240000p6q4r414314400
381{1,1,6,7}3265920p7q6rs15425920
382{1,4,13}3317760p13q4r18351840
383{1,1,2,3,6}3326400p6q3r2st1358640
384{8,9}3359232p9q817232832
385{1,1,1,15}3440640p15qrs18481408
386{3,3,10}3456000p10q3r316315360
387{3,17}3538944p17q320282944
388{1,1,1,2,10}3548160p10q2rst1559216
389{1,1,1,2,2,4}3603600p4q2r2stu11655440
390{1,2,4,8}3628800p8q4r2s15425200
391{2,2,14}3686400p14q2r218357600
392{1,6,10}3732480p10q6r17346080
393{1,1,1,1,1,8}3843840p8qrstu136111360
394{1,1,3,12}3870720p12q3rs17426880
395{1,2,2,2,5}3880800p5q2r2s2t12510080
396{3,5,7}3888000p7q5r31537200
397{1,1,18}3932160p18qr20344160
398{5,14}3981312p14q519249152
399{1,1,1,4,7}3991680p7q4rst14510368
400{1,1,1,1,1,1,4}4084080p4qrstuv10792400
401{2,4,11}4147200p11q4r217363120
402{1,1,3,3,4}4158000p4q3r3st12510800
403{22}4194304p2222136304

 

 

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