## Friday, July 28

I've just created a new sequence of pandigital factorization integers (PFIs). My interest here is the possibility of finding long cascades of these numbers where the difference between adjacent terms is also pandigital. Here's an example:

0.                  2*3*107*45869
1.   + 1045968723 = 3*6047*59281
2.   + 1369754820 = 3*97*8402651
3.   + 1064358927 = 2^8*3*4569701
4.   + 1802467395 = 3*47^2*801569
5.   + 4987621053 = 2^9*3*6705481
6.   + 2480359671 = 3*71^2*845069
7.   + 3154087296 = 3*89^2*670541
8.   + 9241038675 = 2*3*7^5*41*6089

The numbers in the right-most column are the PFIs and the eight additions show the pandigital differences between adjacent terms. In this particular instance the differences are the only way to bridge 2*3*107*45869 to 2*3*7^5*41*6089 but many such cascades have alternate routes to get from the smallest to the largest term.

I only started searching this morning but I already have a 48-term cascade starting with 3^2*5*7*108649 and ending with 3^8*19^2*65407 that only has one minor path variant:

0.                  3^2*5*7*108649
1.   + 1235068974 = 3^4*15670289
2.   + 1047635289 = 2*3^5*7*681049
3.   + 1380467529 = 3^4*7*6520981
4.   + 1067428935 = 2*3^5*9804167
5.   + 1365278409 = 3^4*75680291
6.   + 1063974852 = 3^5*7^2*604189
7.   + 1375960284 = 3^6*419*28057
8.   + 1457863920 = 3^5*41267089
9.   + 1786520934 = 3^5*48619027
10.   + 2098416375 = 2^8*3^4*670951
11.   + 1643978025 = 3^5*64019827
12.   + 1983275604 = 3^6*5*4812097
13.   + 2019647358 = 3^5*80492761
14.   + 1348657290 = 3^5*86042791
15.   + 1563289470 = 3^5*92476081
16.   + 1372460598 = 3^5*98124067
17.   + 1269784350 = 3^6*71*485209
18.   + 1658423970 = 3^6*521*70489
19.   + 2186759430 = 3^6*79*502841
20.   + 1079682534 = 3^6*5*8241097
21.   + 1364529078 = 3^8*29*165047
22.   + 1276894530 = 3^6*487*92051
23.   + 3056148792 = 3^6*5*9804217
24.   + 4069512738 = 3^6*941*58027
25.   + 5178390264 = 3^7*20568941
26.   + 2416753098 = 3^8*5*7*206419
27.   + 6429583170 = 3^8*5*1640927
28.   + 8409356172 = 3^7*28459061
29.   + 1285496730 = 3^7*29046851
30.   + 3750219486 = 3^7*61*504289
31.   + 3671284095 = 2*3^8*5406719
32.   + 8136420759 = 3^8*257*46901
33.   + 4186029537 = 2*3^8*7*906541
34.   + 5704196823 = 3^7*40682951
35.   + 6095824371 = 2^8*3^6*509417
36.   + 7459630281 = 3^8*15627049
37.   + 6319542078 = 3^8*16590247
38.   + 1907234586 = 3^7*50642819
39.   + 3275841690 = 3^7*52140689
40.   + 4807325619 = 2*3^8*9056471
41.   + 3754269810 = 2^5*3^10*64879
42.   + 9270581463 = 3^8*5*4019627
43.   + 5681970342 = 3^8*20964157
44.   + 2617948350 = 3^7*64089521
45.   + 4139728560 = 3^7*65982401
46.   + 9045318276 = 3^8*7*641*5209
47.   + 1569023784 = 3^8*19^2*65407

The variant:

44.   + 4016985372 = 3^8*21576409
45.   + 2740691538 = 3^7*65982401