Runs of Identical Digits in Cubed Numbers

The number associated with my diurnal age yesterday, 26942, has the property that its cube contains a run of five 8's since$$26942^3=19556426288888$$Such a property is not that common. In the range of numbers up to one million, there are only 514 such numbers. They are listed below. The digit sequence "942" occurs quite frequently the numbers containing this digit sequence have been marked in bold (permalink).

NUMBERS WHOSE CUBES CONTAIN THE DIGIT SEQUENCE 88888 (five 8's)

1942, 9615, 19420, 26942, 27402, 32456, 51942, 56676, 57768, 76942, 87455, 88097, 96150, 101942, 105639, 115315, 126942, 134727, 136421, 142578, 143504, 151942, 152605, 161796, 176942, 182605, 183004, 194200, 201942, 226942, 228234, 231627, 242355, 251942, 266319, 269420, 274020, 276942, 283928, 289364, 295827, 301942, 324560, 325776, 326942, 329844, 351942, 360598, 376942, 377881, 389055, 391485, 395027, 401942, 426942, 432304, 446288, 449611, 451942, 466836, 476942, 481869, 481892, 501942, 503767, 506209, 512852, 513843, 517872, 517973, 519112, 519420, 519918, 526942, 539523, 543887, 547416, 551942, 566760, 573766, 576942, 577680, 578113, 580208, 593342, 596116, 596169, 601942, 602507, 602663, 620486, 624538, 626428, 626942, 642355, 651942, 653484, 661064, 675389, 676942, 682768, 684018, 701942, 710646, 711529, 723609, 726942, 751942, 758886, 761096, 761361, 769420, 776942, 787777, 788225, 789482, 791466, 801942, 825672, 826942, 830088, 836354, 851942, 852004, 855117, 861724, 871089, 874550, 876942, 877538, 879792, 880970, 883181, 899467, 901942, 908223, 908256, 915429, 921383, 926942, 943483, 948083, 950254, 951942, 961497, 961498, 961499, 961500, 976942, 980221, 985426, 985521, 987095, 993312

If we extend the run to six 8's, the following numbers qualify in the range up to one million. Note the digit sequence 6942 predominates (permalink).

NUMBERS WHOSE CUBES CONTAIN THE DIGIT SEQUENCE 888888 (six 8's)

76942 --> 455502130888888

326942 --> 34947180653888888

360598 --> 46888888826167192

576942 --> 192042109176888888

769420 --> 455502130888888000

826942 --> 565490287699888888

830088 --> 571968888883241472

877538 --> 675768268888884872

883181 --> 688888844916818741

948083 --> 852195188888887787

985426 --> 956912108888888776

Trying for seven 8's, only these numbers qualify (permalink):

NUMBERS WHOSE CUBES CONTAIN THE DIGIT SEQUENCE 8888888 (seven 8's)

360598 --> 46888888826167192

948083 --> 852195188888887787

985426 --> 956912108888888776

There are no number in the range up to one million (and even ten million) that produce eight 8's when cubed. So what about seven 7's. Are there numbers in the range up to one million, that produce seven 7's when cubed? It turns out that there are not. However, there are a few numbers that produce six 6's (permalink).

NUMBERS WHOSE CUBES CONTAIN THE DIGIT SEQUENCE 666666 (six 6's)

40548 --> 66666601046592

159238 --> 4037756666669272

205408 --> 8666666043277312

266151 --> 18853166666640951

405480 --> 66666601046592000

480108 --> 110666666397419712

537361 --> 155166666621176881

698474 --> 340761666666040424

699637 --> 342466666667067853

705579 --> 351266666653819539

827515 --> 566666611464815875

892954 --> 712011914706666664

There is only one number that produces seven 6's and that is 699637 (marked in bold above). The following numbers produce six 5's (permalink):

NUMBERS WHOSE CUBES CONTAIN THE DIGIT SEQUENCE 555555 (five 5's)

67354 --> 305555550877864

269416 --> 19555555256183296

516602 --> 137869515555559208

673540 --> 305555550877864000

981344 --> 945069645855555584

There are three numbers whose cubes contain seven 4's (permalink):

NUMBERS WHOSE CUBES CONTAIN THE DIGIT SEQUENCE 4444444 (seven 4's)

134708 --> 2444444407022912

720993 --> 374794444444986657

822389 --> 556201144444449869

There are 11 numbers whose cubes produce six 3's:

NUMBERS WHOSE CUBES CONTAIN THE DIGIT SEQUENCE 333333 (six 6's)

32183 --> 33333397072487

321830 --> 33333397072487000

417367 --> 72703333334369863

446477 --> 89001489338333333

510873 --> 133333368622708617

532302 --> 150825333333127608

632748 --> 253333336694092992

664437 --> 293333339149701453

675071 --> 307643933333382911

689701 --> 328082123333339101

767615 --> 452303926733333375

There are two numbers that produce seven 2's when cubed.

NUMBERS WHOSE CUBES CONTAIN THE DIGIT SEQUENCE 2222222 (seven 2's)

339247 --> 39043437522222223

475741 --> 107674222222294021

The following numbers produce six 1's:

NUMBERS WHOSE CUBES CONTAIN THE DIGIT SEQUENCE 111111 (six 1's)

48075 --> 111111210421875

194967 --> 7411111162029063

271234 --> 19954111111440904

288471 --> 24005263647111111

480750 --> 111111210421875000

480751 --> 111111903785004751

567232 --> 182508110911111168

758847 --> 436981111119711423

768373 --> 453645165661111117

790054 --> 493140111111077464

857466 --> 630450111111270696

915441 --> 767169057611111121

923058 --> 786478711961111112

926419 --> 795101111114018059

There are two numbers that produce a run of seven 9's:

NUMBERS WHOSE CUBES CONTAIN THE DIGIT SEQUENCE 9999999 (seven 9's)

764149 --> 446204706999999949

887904 --> 699999995878539264

For zeroes, things are relatively trivial so I won't bother with that. Again, there is a lot more to be explored here. For example, what about runs of consecutive digits? 177434 when cubed produces a run of consecutive digits from 1 to 8: 5586123456782504.