A340695
a(n) is the next perfect power after the earliest occurrence of n consecutive perfect powers, all of which are squares with exponents equal to 2.
Original entry on oeis.org
8, 64, 216, 512, 1728, 4913, 12167, 19683, 32768, 74088, 148877, 175616, 328509, 493039, 753571, 1259712, 1860867, 2406104, 3375000, 4330747, 5929741, 8365427, 11089567, 13824000, 18191447, 23639903, 28934443, 36264691, 43614208, 53582633, 67917312, 81182737, 97972181
Offset: 1
See A340661.
From _David A. Corneth_, Jan 19 2021: (Start)
a(3) = 216 as in the perfect powers we see ..., 128 = 2^7, 144 = 12^2, 169 = 13^2, 196 = 14^2, 216 = 6^3, ... . We write them as powers of m^k where k is chosen as large as possible such that m and k are integers.
Then between two perfect powers with k > 2 (being 128 = 2^7 and 216 = 6^3) we have three consecutive perfect powers with k = 2. As 216 closes this earliest streak of 3, a(3) = 216. (End)
A340661
a(n) is the start of the first occurrence of n consecutive perfect powers, all of which are squares with exponents equal to 2 (A111245).
Original entry on oeis.org
4, 36, 144, 361, 1369, 4225, 10816, 17689, 29929, 69169, 140625, 166464, 314721, 474721, 729316, 1225449, 1817104, 2353156, 3308761, 4251844, 5832225, 8242641, 10942864, 13653025, 17986081, 23396569, 28654609, 35940025, 43243776, 53158681, 67420521, 80622441, 97337956
Offset: 1
Table of initial terms of a(n), A340662, A340663, A340664, and A340695:
bounded below by n consecutive squares terminated by
| a(n) A340662(n) A340695(n)
n | | A340663(n)^2 | A340664(n)^2 |
1 1 = 1^(>2), 4 = 2^2 4 = 2^2, 8 = 2^ 3
2 32 = 2^ 5, 36 = 6^2 ... 49 = 7^2, 64 = 2^ 6
3 128 = 2^ 7, 144 = 12^2 ... 196 = 14^2, 216 = 6^ 3
4 343 = 7^ 3, 361 = 19^2 ... 484 = 22^2, 512 = 2^ 9
5 1331 = 11^ 3, 1369 = 37^2 ... 1681 = 41^2, 1728 = 12^ 3
6 4096 = 2^12, 4225 = 65^2 ... 4900 = 70^2, 4913 = 17^ 3
7 10648 = 22^ 3, 10816 = 104^2 ... 12100 = 110^2, 12167 = 23^ 3
8 17576 = 26^ 3, 17689 = 133^2 ... 19600 = 140^2, 19683 = 3^ 9
9 29791 = 31^ 3, 29929 = 173^2 ... 32761 = 181^2, 32768 = 2^15
A340662
a(n) is the end of the first occurrence of n consecutive perfect powers, all of which are squares with exponents equal to 2 (A111245).
Original entry on oeis.org
4, 49, 196, 484, 1681, 4900, 12100, 19600, 32761, 73984, 148225, 175561, 328329, 492804, 753424, 1258884, 1860496, 2405601, 3374569, 4330561, 5929225, 8363664, 11088900, 13823524, 18190225, 23639044, 28933641, 36264484, 43612816, 53582400, 67914081, 81180100, 97970404
Offset: 1
A340664
a(n)^2 is the end of the first occurrence of n consecutive perfect powers, all of which are squares with exponents equal to 2 (A111245).
Original entry on oeis.org
2, 7, 14, 22, 41, 70, 110, 140, 181, 272, 385, 419, 573, 702, 868, 1122, 1364, 1551, 1837, 2081, 2435, 2892, 3330, 3718, 4265, 4862, 5379, 6022, 6604, 7320, 8241, 9010, 9898, 10913, 12029, 12969, 14149, 15363, 16610, 17928, 19199, 20704, 22331, 23912, 25794, 27406
Offset: 1
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