Problem of the Month (February 2018)

Define the prime signature of a positive integer n to be the exponents to which the prime factors of n are raised. For two prime signatures A and B, what is the smallest number n with prime signature A where the prime signature of n+1 is B?

ANSWERS

George Sicherman, Joe DeVincentis, Gordon Atkinson, and Lewis Chen contributed solutions.

n \ n+1 1 2 11 3 21 111 4 31 22 211 1111 5 41 32 311 221 2111 11111
1 2 3 5 7 11 29 none 23 none 59 389 31 47 71 167 179 419 2309
2 4 none 9 none 49 169 none none none 1849 2209 none 57,121 none 37,249 87,784,
138,523,761 24,649 214,369
11 6 none 14 26 51 65 15 39 35 155 209 none 111 391 119 299 629 3569
3 none 8 none none 27 2197 none 343 none 125 50,653 none none none 3,442,951 1331 24,389 1,295,029
21 12 none 20 124 44 153 none 188 99 275 909 242 175 1124 279 1475 1859 16,925
111 30 none 105 205,378 170 230 none 231 195 494 1869 16,806 890 3266 455 1547 1595 10,373
4 16 none 81 none none 130,321 none none none 1,026,625,681 3,418,801 none none none 2,484,810,622,
489,147,681 none 328,385,156,401 21,517,662,721
31 40 24 54 none 152 104 none 135 none 459 1784 none 1647 44216 2456 3699 1715 22,329
22 36 none 225 none 1444 441 none 69,189,124 none 8649 15,129 none 96,393,124 ? 3249 1,643,524 106,929 199,809
211 60 none 84 342 260 644 none 350 2115 819 1364 3124 495 15124 1287 2204 4059 17,289
1111 210 none 390 1330 930 714 none 1430 1155 2870 7314 147,008,442 2415 27435 3255 6699 10,659 40,754
5 none none 32 none 243 3125 none none none 161,051 371,293 none none none 16,807 ? 1,350,125,107 2,073,071,593
41 112 48 176 4912 368 272 80 567 6723 1424 4304 1,419,856 2511 3,474,247 944 3824 1539 10,625
32 72 none 200 none 1323 968 none 2888 675 3267 21,125 none 3087 ? 171,125 344,763 33,275 830,297
311 270 120 264 6858 1674 594 none 1160 440 1880 4185 133,923,204,
398,129,542 3320 122,824 2295 19,899 9944 21,944
221 180 none 252 2196 980 1044 none 1575 51,075 3380 2925 4,928,488,013,
421,691,156 10,575 793,116 16,820 23,275 18,675 48,069
2111 420 none 780 24,388 924 2820 none 5124 13,923 4004 9009 2,476,098 4940 54,924 9044 22,724 11,780 66,044
11111 2310 none 3990 1,225,042 19,635 10,010 none 10,374 64,515 27,390 38,570 229,345,006 15,470 336,090 30,590 119,510 85,085 378,014
6 none none 64 none none 729 none none none ? 1,771,561 none none none none none 117,649 148,035,889
51 96 none 160 none 2528 608 none 4832 224 1952 2144 none 1376 240,736 3104 51,424 10,449 39,584
42 400 none 144 none 324 3969 none 4624 none 9801 58,564 none 358,420,624 ? 2,424,249 71,824 257,049 866,761
411 240 528 816 670,611,173,776 1232 560 624 3536 1520 2960 4784 9,033,525,
579,302,992 23,247 1,723,024 6992 18,512 15,344 36,464
33 none none 216 none none 1000 none ? none 2744 35,937 none 3375 ? 59,319 405,224 54,872 3,048,625
321 540 360 504 2,588,282,116 3500 2232 none 6615 1224 6174 11,304 115,856,200 16,335 219,500 15,092 111,475 27,675 156,584
3111 1320 840 1560 2,048,382 8360 2184 none 11,960 4760 10,395 11,934 78,502,725,750 29,295 1,773,592 19,250 73,899 51,128 140,504
222 4356 none 900 none 4900 11,025 none 101,124 none 127,449 65,025 none 33,124 465,124 2,082,249 1,249,924 915,849 3,822,025
2211 2340 none 1260 3,869,892 7380 5796 none 26,775 857,475 7524 30,380 1,408,514,752,348 32,175 1,0754,550 20,124 67,275 79,100 111,825
21111 4620 none 5460 2,685,618 11,220 13,650 none 81,620 75,075 28,860 90,804 28,629,150 98,124 4,121,740 107,415 162,260 136,850 254,540
111111 43,890 none 30,030 80,062,990 124,410 53,130 none 255,255 2,390,115 184,470 393,414 714,924,298 983,535 50,828,946 569,415 1,103,102 447,810 2,162,094

n \ n+1	1	2	11	3	21	111	4	31	22	211	1111	5	41	32	311	221	2111	11111
1	2	3	5	7	11	29	none	23	none	59	389	31	47	71	167	179	419	2309
2	4	none	9	none	49	169	none	none	none	1849	2209	none	57,121	none	37,249	87,784, 138,523,761	24,649	214,369
11	6	none	14	26	51	65	15	39	35	155	209	none	111	391	119	299	629	3569
3	none	8	none	none	27	2197	none	343	none	125	50,653	none	none	none	3,442,951	1331	24,389	1,295,029
21	12	none	20	124	44	153	none	188	99	275	909	242	175	1124	279	1475	1859	16,925
111	30	none	105	205,378	170	230	none	231	195	494	1869	16,806	890	3266	455	1547	1595	10,373
4	16	none	81	none	none	130,321	none	none	none	1,026,625,681	3,418,801	none	none	none	2,484,810,622, 489,147,681	none	328,385,156,401	21,517,662,721
31	40	24	54	none	152	104	none	135	none	459	1784	none	1647	44216	2456	3699	1715	22,329
22	36	none	225	none	1444	441	none	69,189,124	none	8649	15,129	none	96,393,124	?	3249	1,643,524	106,929	199,809
211	60	none	84	342	260	644	none	350	2115	819	1364	3124	495	15124	1287	2204	4059	17,289
1111	210	none	390	1330	930	714	none	1430	1155	2870	7314	147,008,442	2415	27435	3255	6699	10,659	40,754
5	none	none	32	none	243	3125	none	none	none	161,051	371,293	none	none	none	16,807	?	1,350,125,107	2,073,071,593
41	112	48	176	4912	368	272	80	567	6723	1424	4304	1,419,856	2511	3,474,247	944	3824	1539	10,625
32	72	none	200	none	1323	968	none	2888	675	3267	21,125	none	3087	?	171,125	344,763	33,275	830,297
311	270	120	264	6858	1674	594	none	1160	440	1880	4185	133,923,204, 398,129,542	3320	122,824	2295	19,899	9944	21,944
221	180	none	252	2196	980	1044	none	1575	51,075	3380	2925	4,928,488,013, 421,691,156	10,575	793,116	16,820	23,275	18,675	48,069
2111	420	none	780	24,388	924	2820	none	5124	13,923	4004	9009	2,476,098	4940	54,924	9044	22,724	11,780	66,044
11111	2310	none	3990	1,225,042	19,635	10,010	none	10,374	64,515	27,390	38,570	229,345,006	15,470	336,090	30,590	119,510	85,085	378,014
6	none	none	64	none	none	729	none	none	none	?	1,771,561	none	none	none	none	none	117,649	148,035,889
51	96	none	160	none	2528	608	none	4832	224	1952	2144	none	1376	240,736	3104	51,424	10,449	39,584
42	400	none	144	none	324	3969	none	4624	none	9801	58,564	none	358,420,624	?	2,424,249	71,824	257,049	866,761
411	240	528	816	670,611,173,776	1232	560	624	3536	1520	2960	4784	9,033,525, 579,302,992	23,247	1,723,024	6992	18,512	15,344	36,464
33	none	none	216	none	none	1000	none	?	none	2744	35,937	none	3375	?	59,319	405,224	54,872	3,048,625
321	540	360	504	2,588,282,116	3500	2232	none	6615	1224	6174	11,304	115,856,200	16,335	219,500	15,092	111,475	27,675	156,584
3111	1320	840	1560	2,048,382	8360	2184	none	11,960	4760	10,395	11,934	78,502,725,750	29,295	1,773,592	19,250	73,899	51,128	140,504
222	4356	none	900	none	4900	11,025	none	101,124	none	127,449	65,025	none	33,124	465,124	2,082,249	1,249,924	915,849	3,822,025
2211	2340	none	1260	3,869,892	7380	5796	none	26,775	857,475	7524	30,380	1,408,514,752,348	32,175	1,0754,550	20,124	67,275	79,100	111,825
21111	4620	none	5460	2,685,618	11,220	13,650	none	81,620	75,075	28,860	90,804	28,629,150	98,124	4,121,740	107,415	162,260	136,850	254,540
111111	43,890	none	30,030	80,062,990	124,410	53,130	none	255,255	2,390,115	184,470	393,414	714,924,298	983,535	50,828,946	569,415	1,103,102	447,810	2,162,094

n \ n+1 6 51 42 411 33 321 3111 222 2211 21111 111111
1 none 223 none 239 none 359 839 none 1259 10,709 30,029
2 none ? none 196249 none ? 94,249 none 20,187,049 2,582,449 8,508,889
11 none 95 143 335 215 755 2183 899 2771 4619 39,269
3 none none none 148,877 none 6859 4913 none 571,787 22,665,187 51,895,117
21 63 927 none 1359 42,216,896,167 539 3479 none 3275 6929 229,709
111 none 1311 399 1455 3374 935 5015 4355 3059 14,819 111,929
4 none ? none ? none ? ? none ? 6,216,250,684,081 7,538,588,431,201
31 none 351 783 4239 999 9125 5319 none 9099 57,875 230,229
22 none 53,553,124 none 651,249 none 19,740,249 1,117,249 none 28,697,449 986,049 22,534,009
211 none 735 1935 4655 9260 4599 7175 30,275 14,651 43,757 149,225
1111 none 4130 8463 7215 42,874 17,835 17,765 34,595 13,299 60605 255,254
5 none none none 229,345,007 none 1,804,229,351 418,195,493 none 2,476,099 1,419,857 115,856,201
41 none 2672 none 19,375 245,314,375 14,499 16,119 none 47,024 101,744 562,869
32 none 41,503 none 36,125 none 1,431,125 630,125 none 186,003 224,939 23,437,829
311 728 5535 2024 6831 185,192 11,799 19,304 27,224 31,899 58,904 399,125
221 none 1700 ? 17,775 10,647 82,124 34,839 28,899 94,275 211,925 1,824,129
2111 none 8415 5775 12,495 132,650 29,799 27,404 23,715 56,924 83,979 424,269
11111 none 41,055 126,735 113,295 658,502 114,855 58,695 81,795 274,274 310,155 1,774,409
6 none none none 186,374,
892,382,561 none none 279,501,644,934,
771,052,249 none ? 4,750,104,241 10,779,215,329
51 none 8991 58,563 11,744 35,936 14,624 26,487 none 46,899 126,944 1,349,984
42 none 900,471,940,624 none 15,460,624 none 544,744,372,624 22,024,249 none 9,865,881 1,306,449 27,910,089
411 none 66,015 178,928 5264 9,129,328 39,824 91,664 11,024 79,299 108,459 672,944
33 none ? none 17,576 none 42,875 238,328 none 8,365,427 274,625 55,306,341
321 none 48,735 1,263,375 24,975 3,525,688,647 41,624 12,375 38,024 105,875 248,675 982,904
3111 none 16,280 29,240 63,855 97,335 58,995 21,735 81,224 134,504 122,264 1,269,729
222 none 11,517,153,124 none 44,796,249 none 1,762,488,324 23,107,249 none 1,979,649 19,421,649 4,678,569
2211 none 47,775 364,815 125,775 614,124 75,140 195,975 407,043 223,244 241,604 2,804,229
21111 none 319,332 583,695 69,615 1,191,015 399,399 169,575 244,035 467,324 1,241,954 2,964,884
111111 none 310,310 4,145,295 899,990 24,642,170 1,225,070 1,565,655 1,716,099 2,146,130 3,139,730 11,243,154

n \ n+1	6	51	42	411	33	321	3111	222	2211	21111	111111
1	none	223	none	239	none	359	839	none	1259	10,709	30,029
2	none	?	none	196249	none	?	94,249	none	20,187,049	2,582,449	8,508,889
11	none	95	143	335	215	755	2183	899	2771	4619	39,269
3	none	none	none	148,877	none	6859	4913	none	571,787	22,665,187	51,895,117
21	63	927	none	1359	42,216,896,167	539	3479	none	3275	6929	229,709
111	none	1311	399	1455	3374	935	5015	4355	3059	14,819	111,929
4	none	?	none	?	none	?	?	none	?	6,216,250,684,081	7,538,588,431,201
31	none	351	783	4239	999	9125	5319	none	9099	57,875	230,229
22	none	53,553,124	none	651,249	none	19,740,249	1,117,249	none	28,697,449	986,049	22,534,009
211	none	735	1935	4655	9260	4599	7175	30,275	14,651	43,757	149,225
1111	none	4130	8463	7215	42,874	17,835	17,765	34,595	13,299	60605	255,254
5	none	none	none	229,345,007	none	1,804,229,351	418,195,493	none	2,476,099	1,419,857	115,856,201
41	none	2672	none	19,375	245,314,375	14,499	16,119	none	47,024	101,744	562,869
32	none	41,503	none	36,125	none	1,431,125	630,125	none	186,003	224,939	23,437,829
311	728	5535	2024	6831	185,192	11,799	19,304	27,224	31,899	58,904	399,125
221	none	1700	?	17,775	10,647	82,124	34,839	28,899	94,275	211,925	1,824,129
2111	none	8415	5775	12,495	132,650	29,799	27,404	23,715	56,924	83,979	424,269
11111	none	41,055	126,735	113,295	658,502	114,855	58,695	81,795	274,274	310,155	1,774,409
6	none	none	none	186,374, 892,382,561	none	none	279,501,644,934, 771,052,249	none	?	4,750,104,241	10,779,215,329
51	none	8991	58,563	11,744	35,936	14,624	26,487	none	46,899	126,944	1,349,984
42	none	900,471,940,624	none	15,460,624	none	544,744,372,624	22,024,249	none	9,865,881	1,306,449	27,910,089
411	none	66,015	178,928	5264	9,129,328	39,824	91,664	11,024	79,299	108,459	672,944
33	none	?	none	17,576	none	42,875	238,328	none	8,365,427	274,625	55,306,341
321	none	48,735	1,263,375	24,975	3,525,688,647	41,624	12,375	38,024	105,875	248,675	982,904
3111	none	16,280	29,240	63,855	97,335	58,995	21,735	81,224	134,504	122,264	1,269,729
222	none	11,517,153,124	none	44,796,249	none	1,762,488,324	23,107,249	none	1,979,649	19,421,649	4,678,569
2211	none	47,775	364,815	125,775	614,124	75,140	195,975	407,043	223,244	241,604	2,804,229
21111	none	319,332	583,695	69,615	1,191,015	399,399	169,575	244,035	467,324	1,241,954	2,964,884
111111	none	310,310	4,145,295	899,990	24,642,170	1,225,070	1,565,655	1,716,099	2,146,130	3,139,730	11,243,154

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