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Table 3 First digit distribution of powerful integer powers up to \( 10^{s \cdot m} , \, m = 4, \ldots ,10 \)

From: A first digit theorem for powerful integer powers

s = 1/1st digit 21,044 67,231 214,122 680,330 2,158,391 6,840,384 21,663,503 68,575,557
1 4007 12,822 40,904 130,091 412,967 1,309,267 4,147,585 13,131,523
2 3083 9871 31,450 99,940 317,119 1,005,145 3,183,613 10,078,415
3 2610 8331 26,524 84,296 267,441 847,620 2,684,438 8,497,665
4 2300 7347 23,395 74,293 235,677 746,893 2,365,308 7,487,199
5 2073 6636 21,136 67,171 213,092 675,309 2,138,598 6,769,360
6 1913 6110 19,454 61,808 196,043 621,153 1,966,875 6,225,554
7 1786 5696 18,119 57,541 182,478 578,178 1,830,822 5,794,823
8 1682 5363 17,044 54,078 171,442 543,121 1,719,686 5,442,855
9 1590 5055 16,096 51,112 162,132 513,698 1,626,578 5,148,163
s = 2/1st digit 21,044 67,231 214,122 680,330 2,158,391 6,840,384 21,663,503 68,575,557
1 5101 16,305 51,985 165,261 524,441 1,662,341 5,265,235 16,668,479
2 3423 10,953 34,878 110,838 351,732 1,114,820 3,530,987 11,177,711
3 2653 8468 26,991 85,800 272,191 862,616 2,731,775 8,647,291
4 2194 7011 22,334 70,939 225,030 713,170 2,258,528 7,149,260
5 1882 6029 19,196 60,981 193,455 613,024 1,941,382 6,145,104
6 1662 5320 16,935 53,789 170,584 540,582 1,711,903 5,418,705
7 1506 4784 15,209 48,289 153,178 485,450 1,537,240 4,865,873
8 1360 4348 13,848 43,989 139,490 441,889 1,399,282 4,429,024
9 1263 4013 12,746 40,444 128,290 406,492 1,287,171 4,074,110
s = 3/1st digit 21,044 67,231 214,122 680,330 2,158,391 6,840,384 21,663,503 68,575,557
1 5506 17,585 56,011 178,000 564,823 1,790,255 5,670,207 17,949,938
2 3521 11,272 35,904 114,105 362,030 1,147,413 3,633,936 11,503,374
3 2646 8467 26,989 85,776 272,126 862,442 2,731,429 8,646,513
4 2148 6855 21,840 69,434 220,301 698,158 2,210,950 6,998,516
5 1818 5817 18,502 58,717 186,183 590,041 1,868,635 5,914,789
6 1583 5046 16,076 51,051 161,930 513,041 1,624,701 5,142,951
7 1403 4473 14,233 45,264 143,639 455,234 1,441,644 4,563,270
8 1254 4032 12,857 40,802 129,418 410,055 1,298,522 4,110,259
9 1164 3683 11,709 37,180 117,940 373,744 1,183,478 3,745,946
s = 4/1st digit 21,044 67,231 214,122 680,330 2,158,391 6,840,384 21,663,503 68,575,557
1 5698 18,216 58,062 184,559 585,590 1,855,943 5,878,103 18,607,799
2 3577 11,427 36,392 115,638 366,864 1,162,715 3,682,401 11,656,783
3 2656 8466 26,970 85,665 271,811 861,509 2,728,356 8,636,596
4 2110 6777 21,591 68,632 217,705 689,893 2,184,938 6,916,359
5 1790 5702 18,115 57,504 182,437 578,178 1,831,099 5,796,212
6 1528 4891 15,615 49,655 157,550 499,308 1,581,133 5,004,805
7 1350 4335 13,793 43,814 138,938 440,298 1,394,424 4,413,972
8 1226 3888 12,354 39,236 124,518 394,542 1,249,524 3,955,235
9 1109 3529 11,230 35,627 112,978 357,998 1,133,525 3,587,796
s = 5/1st digit 21,044 67,231 214,122 680,330 2,158,391 6,840,384 21,663,503 68,575,557
1 5809 18,610 59,329 188,524 598,205 1,895,921 6,004,544 19,007,822
2 3609 11,527 36,670 116,505 369,658 1,171,550 3,710,470 11,745,615
3 2640 8451 26,948 85,615 271,535 860,564 2,725,421 8,627,359
4 2126 6756 21,449 68,139 216,096 684,815 2,168,758 6,865,158
5 1738 5591 17,871 56,780 180,199 571,066 1,808,411 5,724,220
6 1509 4821 15,356 48,817 154,913 490,922 1,554,819 4,921,783
7 1323 4248 13,510 42,921 136,137 431,469 1,366,413 4,325,270
8 1202 3805 12,076 38,352 121,654 385,457 1,220,575 3,863,552
9 1088 3422 10,913 34,677 109,994 348,620 1,104,092 3,494,778
s → ∞/1st digit 21,044 67,231 214,122 680,330 2,158,391 6,840,384 21,663,503 68,575,557
1 6335 20,239 64,457 204,800 649,740 2,059,161 6,521,364 20,643,300
2 3706 11,839 37,705 119,800 380,074 1,204,532 3,814,754 12,075,556
3 2629 8400 26,752 85,000 269,667 854,629 2,706,611 8,567,743
4 2039 6515 20,751 65,931 209,170 662,902 2,099,410 6,645,658
5 1666 5323 16,954 53,869 170,904 541,630 1,715,343 5,429,898
6 1409 4501 14,335 45,546 144,497 457,942 1,450,302 4,590,913
7 1220 3899 12,417 39,454 125,169 396,687 1,256,309 3,976,830
8 1076 3439 10,953 34,801 110,407 349,903 1,108,143 3,507,813
9 963 3076 9798 31,130 98,763 312,999 991,268 3,137,845