cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Views

Author

Hugo Pfoertner, Jan 18 2021

Keywords

Examples

			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

Crossrefs

Cf. A000290, A001597, A111245, A340662, A340663, A340664, A340695.

Formula

a(n) = A340663(n)^2.

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

Views

Author

Hugo Pfoertner, Jan 18 2021

Keywords

Examples

			See A340661.

Crossrefs

Cf. A000290, A001597, A111245, A340661, A340663, A340664, A340695.

Formula

a(n) = A340664(n)^2 = A340663(n)^2 + (n - 1)*(2*A340663(n) + n - 1).

A340663 a(n)^2 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

2, 6, 12, 19, 37, 65, 104, 133, 173, 263, 375, 408, 561, 689, 854, 1107, 1348, 1534, 1819, 2062, 2415, 2871, 3308, 3695, 4241, 4837, 5353, 5995, 6576, 7291, 8211, 8979, 9866, 10880, 11995, 12934, 14113, 15326, 16572, 17889, 19159, 20663, 22289, 23869, 25750, 27361
Offset: 1

Views

Author

Hugo Pfoertner, Jan 18 2021

Keywords

Examples

			See A340661.

Crossrefs

Cf. A000290, A001597, A111245, A340661, A340662, A340664, A340695.

Formula

a(n) = sqrt(A340661(n)).

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

Views

Author

Hugo Pfoertner, Jan 18 2021

Keywords

Examples

			See A340661.

Crossrefs

Cf. A000290, A001597, A111245, A340661, A340662, A340663, A340695.

Formula

a(n) = sqrt(A340662(n)) = A340663(n) + n - 1.

A340642 Perfect powers such that the two immediately adjacent perfect powers both have an exponent greater than 2.

Original entry on oeis.org

4, 9, 25, 225, 676, 2116, 6724, 7921, 8100, 16641, 104329, 131044, 160801, 176400, 372100, 389376, 705600, 4096576, 7306209, 7884864, 47444544, 146385801, 254817369, 373262400, 607622500, 895804900, 1121580100, 1330936324, 1536875209, 2097182025, 2258435529, 2749953600
Offset: 1

Views

Author

Hugo Pfoertner, Jan 14 2021

Keywords

Comments

Apparently, all known terms (checked through 10^18) are squares with maximum exponent 2, i.e., terms of A111245 (squares that are not a higher power). This would imply that of 3 immediately adjacent perfect powers, at least one is a term of A111245. Is there a known counterexample of 3 consecutive perfect powers, none of which is in A111245?

Examples

			The first terms, assuming 1 being at least a cube:
.
  n   p1  x^p1  p2  a(n)  p3  z^p3
                   =y^p2
  1  >2     1   2     4   3     8
  2   3     8   2     9   4    16
  3   4    16   2    25   3    27
  4   3   216   2   225   5   243
  5   4   625   2   676   6   729

Links

Crossrefs

Cf. A001597, A025479, A076467, A097054, A111245, A153158, A340643, A340700, A340701.

Programs

  • PARI
    a340642(limit)={my(p2=999, p1=2, n2=1, n1=4); for(n=5, limit, my(p0=ispower(n)); if(p0>1, if(p2>2&p0>2, print1(n1,", ")); n2=n1; n1=n; p2=p1; p1=p0))};
a340642(50000000)

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

Views

Author

Hugo Pfoertner, Jan 18 2021

Keywords

Comments

The exponent of a(n) is > 2 thus terminating the progression of n consecutive preceding squares with exponents = 2 (A111245).
Is this sequence strictly increasing? - David A. Corneth, Jan 19 2021

Examples

			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)

Links

Crossrefs

Cf. A000290, A001597, A111245, A340661, A340662, A340663, A340664.

Programs

  • PARI
    \\ See Corneth link

A340586 Perfect powers such that the two immediately adjacent perfect powers both have a largest exponent A025479 equal to 2.

Original entry on oeis.org

8, 16, 169, 216, 343, 400, 441, 512, 625, 729, 841, 900, 1156, 1444, 1521, 1600, 1728, 1849, 1936, 2048, 2401, 2601, 2744, 2916, 3125, 3249, 3375, 3600, 3721, 3844, 4096, 4356, 4489, 4624, 4761, 4913, 5184, 5329, 5476, 5625, 5832, 6084, 6241, 6561, 6859, 7056
Offset: 1

Views

Author

Hugo Pfoertner, Jan 14 2021

Keywords

Examples

			a(1) = 8 because its neighboring perfect powers 4 = 2^2 and 9 = 3^2 both have the largest exponent 2.
9 is not in the sequence because both exponents of the neighboring perfect powers 8 = 2^3 and 16 = 2^4 are > 2.
a(2) = 16: neighbors 9 = 3^2 and 25 = 5^2 satisfy the exponent condition.
Next excluded terms: 25 (16 = 2^4, 27 = 3^3), 27 (32 = 2^5), 32 (27 = 3^3), 36 (32 = 2^5), 49 (64 = 2^6), 64 (81 = 3^4), 81 (64 = 2^6), 100 (81 = 3^4), 121 (125 = 5^3), 125 (128 = 2^7), 128 (125 = 5^3), 144 (128 = 2^7).
a(3) = 169: neighbors 144 = 12^2 and 196 = 14^2 satisfy the exponent condition.

Crossrefs

Cf. A000290, A001597, A025479, A111245, A153158, A340587, A340640, A340641.

Programs

  • PARI
    a340586(limit)={my(p2=999,p1=2,n2=1,n1=4);for(n=5,limit,my(p0=ispower(n));if(p0>1,if(p2+p0==4,print1(n1,", "));n2=n1;n1=n;p2=p1;p1=p0))};
a340586(7500)

A340640 Perfect powers such that the two immediately adjacent perfect powers have at least one largest exponent A025479 greater than 2.

Original entry on oeis.org

4, 9, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 196, 225, 243, 256, 289, 324, 361, 484, 529, 576, 676, 784, 961, 1000, 1024, 1089, 1225, 1296, 1331, 1369, 1681, 1764, 2025, 2116, 2187, 2197, 2209, 2304, 2500, 2704, 2809, 3025, 3136, 3364, 3481, 3969
Offset: 1

Views

Author

Hugo Pfoertner, Jan 14 2021

Keywords

Examples

			a(1) = 4 because the next perfect power is 8 = 2^3, i.e., its exponent is > 2.
a(2) = 9: the exponents of the neighbors 8 = 2^3 and 16 = 2^4 are both > 2.
16 is not in the sequence because both neighboring perfect powers 9 = 3^2 and 25 = 5^2 have exponents 2.
Neighbors with exponents > 2 of the next terms: a(3) = 25 (16 = 2^3), a(4) = 27 (32 = 2^5), a(5) = 32 (27 = 3^3), a(6) = 36 (32 = 2^5), a(7) = 49 (64 = 2^6), a(8) = 64 (81 = 3^4).

Crossrefs

Cf. A000290, A001597, A025479, A111245, A153158, A340586, A340641.

Programs

  • PARI
    a340640(limit)={my(p2=999, p1=2, n2=1, n1=4); for(n=5, limit, my(p0=ispower(n)); if(p0>1, if(p2+p0>4, print1(n1, ", ")); n2=n1; n1=n; p2=p1; p1=p0))};
a340640(5000)

A340643 Numbers k such that the two perfect powers immediately adjacent to k^2 both have exponents greater than 2.

Original entry on oeis.org

2, 3, 5, 15, 26, 46, 82, 89, 90, 129, 323, 362, 401, 420, 610, 624, 840, 2024, 2703, 2808, 6888, 12099, 15963, 19320, 24650, 29930, 33490, 36482, 39203, 45795, 47523, 52440, 66050, 69168, 83408, 94248, 94863, 103683, 114284, 164399, 185364, 206442, 222785, 227530, 229180
Offset: 1

Views

Author

Hugo Pfoertner, Jan 14 2021

Keywords

Comments

Within the range of the data, a(n)^2 = A340642(n), i.e., no 3 immediately consecutive perfect powers x^p1, y^p2, z^p3 with min (p1, p2, p3) > 2 are seen. Is there a counterexample?

Links

Crossrefs

Cf. A000290, A001597, A025479, A076467, A097054, A111245, A153158, A340642, A340700, A340701.

Programs

  • PARI
    a340643(limit)={my(p2=999, p1=2, n2=1, n1=4); for(n=5, limit, my(p0=ispower(n)); if(p0>1, if(issquare(n1)&p2>2&p0>2, print1(sqrtint(n1),", ")); n2=n1; n1=n; p2=p1; p1=p0))};
a340643(10^8)
  • PARI
    upto(n) = {n *= n; my(v = List(), res = List([2])); for(i = 2, sqrtnint(n, 3), for(e = 3, logint(n, i), listput(v, i^e) ); ); listsort(v, 1); for(i = 1, #v - 1, if(sqrtint(v[i]) + 1 == sqrtint(v[i+1]) - issquare(v[i+1]), listput(res, sqrtint(v[i+1]-issquare(v[i+1]))); ) ); res }

Extensions

More terms from David A. Corneth, Jan 14 2021
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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