A017913 -id:A017913 - cp's OEIS frontend

A221942 a(n) = floor(sqrt(5*2^n)).

2, 3, 4, 6, 8, 12, 17, 25, 35, 50, 71, 101, 143, 202, 286, 404, 572, 809, 1144, 1619, 2289, 3238, 4579, 6476, 9158, 12952, 18317, 25905, 36635, 51810, 73271, 103621, 146542, 207243, 293085, 414486, 586171, 828972, 1172343, 1657944, 2344687, 3315888, 4689374, 6631776, 9378748, 13263553, 18757497, 26527107

Offset: 0

N. J. A. Sloane, Feb 01 2013

nonn

Theorem 3 of Dubickas implies that infinitely many terms of this sequence are divisible by 2 or 3 (and hence infinitely many composites). - Charles R Greathouse IV, Feb 04 2016

Artūras Dubickas, Prime and composite integers close to powers of a number, Monatsh. Math. 158:3 (2009), pp. 271-284.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A017910, A017913, A017919, A114183, A221718, A221942-A221946.

Floor[Sqrt[5*2^Range[0,50]]] (* Harvey P. Dale, Mar 09 2025 *)

a(n)=sqrtint(5*2^n) \\ Charles R Greathouse IV, Apr 18 2013

A221946 a(n) = floor(sqrt(2*7^n)).

Original entry on oeis.org

1, 3, 9, 26, 69, 183, 485, 1283, 3395, 8983, 23768, 62886, 166380, 440202, 1164665, 3081415, 8152659, 21569910, 57068618, 150989371, 399480328, 1056925602, 2796362297, 7398479214, 19574536080, 51789354498, 137021752562, 362525481486, 959152267937, 2537678370405, 6714065875561, 17763748592841

Offset: 0

N. J. A. Sloane, Feb 01 2013

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A017910, A017913, A017919, A114183, A221718, A221942-A221946.

[Floor(Sqrt(2*7^n)): n in [0..50]]; // G. C. Greubel, Jan 26 2018

Table[Floor[Sqrt[2*7^n]], {n, 0, 50}] (* G. C. Greubel, Jan 26 2018 *)

a(n)=sqrtint(2*7^n) \\ Charles R Greathouse IV, Apr 18 2013

A077626 Largest term in periodic part of continued fraction expansion of square root of 1+3^n or 0 if 1+3^n is square.

Original entry on oeis.org

0, 6, 10, 18, 30, 54, 92, 162, 280, 486, 840, 1458, 2524, 4374, 7574, 13122, 22726, 39366, 68182, 118098, 204550, 354294, 613654, 1062882, 1840964, 3188646, 5522896, 9565938, 16568690, 28697814, 49706070, 86093442, 149118214, 258280326, 447354646, 774840978

Offset: 1

Labos Elemer, Nov 13 2002

nonn

a(n) = 0 iff n = 1, as a consequence of Catalan's conjecture or Mihăilescu's theorem. - Bernard Schott, Apr 25 2022

Cf. A077624-A077635.

Equals 2*A017913(n) for n > 1.

Table[Max[Last[ContinuedFraction[Sqrt[1+3^u]]]], {u, 1, 32}]

a(n) = if (n==1, 0, 2*sqrtint(3^n)); \\ Michel Marcus, Apr 20 2022

a(1) changed and definition clarified by Chai Wah Wu, Sep 18 2021

a(31)-a(36) from Chai Wah Wu, Apr 20 2022

A128444 Array T by antidiagonals: T(n,k)=Floor(k*3^(n/2)).

Original entry on oeis.org

1, 3, 3, 5, 6, 5, 6, 9, 10, 9, 8, 12, 15, 18, 15, 10, 15, 20, 27, 31, 27, 12, 18, 25, 36, 46, 54, 46, 13, 21, 31, 45, 62, 81, 93, 81, 15, 24, 36, 54, 77, 108, 140, 162, 140, 17, 27, 41, 63, 93, 135, 187, 243, 280, 243, 19, 30, 46, 72, 109, 162, 233, 324, 420, 486, 420, 20, 33

Offset: 1

Clark Kimberling, Mar 03 2007

Row 1 = A022838; Row 2 = A008585; Column 1 = A017913; T(n,n) = A128443(n).

			Northwest corner:
1 3 5 6 8
3 6 9 12 15
5 10 15 20 25
9 18 27 36 45

Cf. A128443.

A221944 Floor(sqrt(2*3^n)).

Original entry on oeis.org

1, 2, 4, 7, 12, 22, 38, 66, 114, 198, 343, 595, 1030, 1785, 3092, 5357, 9278, 16071, 27835, 48213, 83507, 144639, 250523, 433919, 751571, 1301759, 2254713, 3905277, 6764139, 11715833, 20292418, 35147500, 60877256, 105442501, 182631769, 316327504, 547895309, 948982513, 1643685929, 2846947541, 4931057788

Offset: 0

N. J. A. Sloane, Feb 01 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A017910, A017913, A017919, A114183, A221718, A221942-A221946.

[Floor(Sqrt(2*3^n)): n in [0..40]]; // Vincenzo Librandi, Mar 11 2014

Floor[Sqrt[2 3^Range[0,40]]] (* Harvey P. Dale, Mar 09 2014 *)

a(n)=sqrtint(2*3^n) \\ Charles R Greathouse IV, Apr 18 2013

A221945 a(n) = floor(sqrt(2*5^n)).

Original entry on oeis.org

1, 3, 7, 15, 35, 79, 176, 395, 883, 1976, 4419, 9882, 22097, 49410, 110485, 247052, 552427, 1235264, 2762135, 6176323, 13810679, 30881617, 69053396, 154408088, 345266983, 772040444, 1726334915, 3860202221, 8631674575, 19301011109, 43158372875, 96505055547, 215791864375, 482525277735

Offset: 0

N. J. A. Sloane, Feb 01 2013

nonn

Floor of norm of (1 + i) * (1 + 2i)^n. - Jon Perry, Dec 06 2013

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A017910, A017913, A017919, A114183, A221718, A221942-A221946.

[Floor(Sqrt(2*5^n)): n in [0..40]]; // G. C. Greubel, Oct 06 2018

Table[Floor[Sqrt[2*5^n]],{n,0,40}] (* Harvey P. Dale, Feb 10 2016 *)

a(n)=sqrtint(2*5^n) \\ Charles R Greathouse IV, Apr 18 2013

A255616 Table read by antidiagonals, T(n,k) = floor(sqrt(k^n)), n >= 0, k >=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 4, 5, 4, 1, 1, 2, 5, 8, 9, 5, 1, 1, 2, 6, 11, 16, 15, 8, 1, 1, 2, 7, 14, 25, 32, 27, 11, 1, 1, 3, 8, 18, 36, 55, 64, 46, 16, 1, 1, 3, 9, 22, 49, 88, 125, 128, 81, 22, 1, 1, 3, 10, 27, 64, 129, 216, 279, 256, 140, 32, 1, 1, 3, 11, 31, 81, 181, 343, 529, 625, 512, 243, 45, 1

Offset: 0

Kival Ngaokrajang, Feb 28 2015

			See table in the links.

G. C. Greubel, Table of n, a(n) for the first 100 antidaigonals, flattened
Kival Ngaokrajang, Example of table T(n,k), n = 0..12, k = 1..10

Rows(1..10): A000012, A000196, A000027, A000093, A000290, A155013, A000578, A155014, A000583, A238170.
Columns(1..10): A000012, A017910, A017913, A000079, A017919, A017922, A017925, A017928, A000244, A017934.
Diagonals: A190343, A066642, A075264.

T[n_, k_] := Floor[Sqrt[k^n]]; Table[T[k, n + 1 - k], {n, 0, 15}, {k, 0, n}] (* G. C. Greubel, Dec 30 2017 *)

{for(i=1,20,for(n=0,i-1,a=floor(sqrt((i-n)^n));print1(a,", ")))}

T(n,k) = floor(sqrt(k^n)), n >= 0, k >=1.

Terms a(81) onward added by G. C. Greubel, Dec 30 2017

A221943 Floor(sqrt(7*2^n)).

Original entry on oeis.org

2, 3, 5, 7, 10, 14, 21, 29, 42, 59, 84, 119, 169, 239, 338, 478, 677, 957, 1354, 1915, 2709, 3831, 5418, 7662, 10836, 15325, 21673, 30651, 43347, 61303, 86695, 122606, 173391, 245213, 346783, 490426, 693567, 980853, 1387135, 1961706, 2774271, 3923412, 5548542, 7846824, 11097085, 15693648, 22194170, 31387297

Offset: 0

N. J. A. Sloane, Feb 01 2013

nonn

Theorem 3 of Dubickas implies that infinitely many terms of this sequence are divisible by 2 or 3 (and hence infinitely many composites). - Charles R Greathouse IV, Feb 04 2016

Artūras Dubickas, Prime and composite integers close to powers of a number, Monatsh. Math. 158:3 (2009), pp. 271-284.

Cf. A017910, A017913, A017919, A114183, A221718, A221942-A221946.

Floor[Sqrt[7*2^Range[0,50]]] (* Harvey P. Dale, Apr 11 2018 *)

a(n)=sqrtint(7<Charles R Greathouse IV, Apr 18 2013

cp's OEIS Frontend

A221942 a(n) = floor(sqrt(5*2^n)).

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A221946 a(n) = floor(sqrt(2*7^n)).

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A077626 Largest term in periodic part of continued fraction expansion of square root of 1+3^n or 0 if 1+3^n is square.

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A128444 Array T by antidiagonals: T(n,k)=Floor(k*3^(n/2)).

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A221944 Floor(sqrt(2*3^n)).

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Magma

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A221945 a(n) = floor(sqrt(2*5^n)).

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Author

Keywords

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Magma

Mathematica

PARI

A255616 Table read by antidiagonals, T(n,k) = floor(sqrt(k^n)), n >= 0, k >=1.

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A221943 Floor(sqrt(7*2^n)).

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Author

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