A017919 -id:A017919 - 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

A017920 Powers of sqrt(5) rounded to nearest integer.

Original entry on oeis.org

1, 2, 5, 11, 25, 56, 125, 280, 625, 1398, 3125, 6988, 15625, 34939, 78125, 174693, 390625, 873464, 1953125, 4367320, 9765625, 21836601, 48828125, 109183007, 244140625, 545915034, 1220703125, 2729575168

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Cf. A017919, A017921.

[Round(Sqrt(5)^n): n in [0..40]]; // Vincenzo Librandi, Nov 19 2011

Floor[(Sqrt[5]^Range[0,40]+1/2)] (* Vincenzo Librandi, Nov 19 2011 *)

a(n)=round(sqrt(5)^n) \\ Charles R Greathouse IV, Nov 18 2011

from math import isqrt
def A017920(n): return (m:=isqrt(k:=5**n))+(k-m*(m+1)>=1) # Chai Wah Wu, Jun 19 2024

A017921 Powers of sqrt(5) rounded up.

Original entry on oeis.org

1, 3, 5, 12, 25, 56, 125, 280, 625, 1398, 3125, 6988, 15625, 34939, 78125, 174693, 390625, 873465, 1953125, 4367321, 9765625, 21836602, 48828125, 109183007, 244140625, 545915034, 1220703125, 2729575168

Offset: 0

N. J. A. Sloane

			sqrt(5)^3 = 11.18033988749895... so a(3) = 12.
sqrt(5)^4 = 25, so a(4) = 25.
sqrt(5)^5 = 55.90169943749474241... so a(5) = 56.

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

Cf. A017919, A017920.

[Ceiling(Sqrt(5^n)): n in [0..40]]; // Vincenzo Librandi, Nov 20 2011

Ceiling[Sqrt[5]^Range[0, 40]] (* Vincenzo Librandi, Nov 20 2011 *)

a(n) = ceil(sqrt(5^n)); \\ Michel Marcus, Dec 14 2017

from math import isqrt
def A017921(n): return isqrt(5**n)+(n&1) # Chai Wah Wu, Jun 19 2024

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

A255216 a(n) = floor((3/sqrt(5))^n).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 18, 25, 34, 45, 61, 82, 110, 147, 198, 266, 357, 479, 642, 862, 1156, 1552, 2082, 2793, 3748, 5028, 6746, 9051, 12143, 16292, 21859, 29327, 39346, 52788, 70823, 95019, 127482, 171035, 229468, 307863, 413042, 554155, 743477, 997479, 1338258

Offset: 0

Kival Ngaokrajang, Feb 25 2015

nonn

a(n) is the total length (rounded down to integer) of the elements of a variant of a 3-element fractal after n iterations, starting with 3 elements, each of whose length is 1/3 (in some units). See illustration in the links.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms

Cf. A017919.

With[{c=3/Sqrt[5]},Table[Floor[c^n],{n,0,50}]] (* Harvey P. Dale, Oct 23 2023 *)

{for(n=0,100,a=floor((3/sqrt(5))^n);print1(a,", "))}

a(n) = floor((3/sqrt(5))^n).

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)).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A017920 Powers of sqrt(5) rounded to nearest integer.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

A017921 Powers of sqrt(5) rounded up.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A255216 a(n) = floor((3/sqrt(5))^n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author