A114183 -id:A114183 - cp's OEIS frontend

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

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

A213912 a(1) = 1; for n>1, a(n) = floor(sqrt(a(n-1))) if that number is not already in the sequence, otherwise a(n) = 3*a(n-1).

Original entry on oeis.org

1, 3, 9, 27, 5, 2, 6, 18, 4, 12, 36, 108, 10, 30, 90, 270, 16, 48, 144, 432, 20, 60, 7, 21, 63, 189, 13, 39, 117, 351, 1053, 32, 96, 288, 864, 29, 87, 261, 783, 2349, 7047, 83, 249, 15, 45, 135, 11, 33, 99, 297, 17, 51, 153, 459, 1377, 37, 111, 333, 999, 31

Offset: 1

Reinhard Zumkeller, Mar 05 2013

Permutation of natural numbers with inverse A213913.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A114183.

a213912 n = a213912_list !! (n-1)
a213912_list = 1 : f [1] where
   f xs@(x:_) = y : f (y : xs) where
     y = if z `notElem` xs then z else 3 * x where z = a000196 x

lista(nn) = {my(k, v=vector(nn)); v[1]=1; for(n=2, nn, if(vecsearch(vecsort(v), k=sqrtint(v[n-1])), v[n]=3*v[n-1], v[n]=k)); v; } \\ Jinyuan Wang, Jun 26 2020

A217727 a(1)=2. For n >= 1, let k = floor(log a(n)). If k >= 2 and k is not already in the sequence then a(n+1)=k, otherwise a(n+1)=a(n)^2.

Original entry on oeis.org

2, 4, 16, 256, 5, 25, 3, 9, 81, 6561, 8, 64, 4096, 16777216, 281474976710656, 33, 1089, 6, 36, 1296, 7, 49, 2401, 5764801, 15, 225, 50625, 10, 100, 10000, 100000000, 18, 324, 104976, 11, 121, 14641, 214358881, 19, 361, 130321, 16983563041, 23, 529, 279841, 12, 144, 20736, 429981696, 184884258895036416, 39

Offset: 1

N. J. A. Sloane, Mar 21 2013

nonn

Does every number >= 2 appear?
Suggested by a posting to the Math Fun Mailing List by Keith F. Lynch, Mar 21 2013.
See A277848 for the index of a given number, i.e., the left inverse of this function. - M. F. Hasler, Nov 20 2016

Alois P. Heinz, Table of n, a(n) for n = 1..662

Cf. A114183.

Digits:=100;
a1:=[2,4,16];
s1:=convert(a1,set);
for n from 3 to 50 do
t1:=a1[n];
k:=floor(log(t1));
if k >= 2 then
   if evalb(k in s1) then a1:=[op(a1),t1^2]; s1:= s1 union {t1^2}
   else a1:=[op(a1),k]; s1 := s1 union {k}; fi;
else
   a1:=[op(a1),t1^2]; s1:= s1 union {t1^2};
fi;
od:
[seq(a1[i],i=1..nops(a1))];

a = {2}; Do[If[And[# >= 2, ! MemberQ[a, #]], AppendTo[a, #], AppendTo[a, a[[n]]^2]] &@ Floor@ Log@ a[[n]], {n, 50}]; a (* Michael De Vlieger, Nov 22 2016 *)

A217727(n,show_all=0,a=2,u=[])={for(i=2,n, show_all&&print1(a","); u=setunion(u, [a]); while(#u>1&&u[2]==u[1]+1, u=u[^1]); my(t=log(a)\1); a=if(t>u[1]&&!setsearch(u, t), t, a^2));a} \\ M. F. Hasler, Nov 22 2016

A277848 Index at which n occurs in A217727, or 0 if there is no such index.

Original entry on oeis.org

1, 7, 2, 5, 18, 21, 11, 8, 28, 35, 46, 74, 54, 25, 3, 129, 32, 39, 78, 58, 133, 43, 62, 6, 95, 91, 152, 171, 87, 71, 190, 16, 148, 328, 19, 181, 167, 51, 227, 405, 618, 355, 344, 254, 440, 83, 926, 22, 67, 277, 100, 157, 1400, 195, 333, 186, 1144, 232, 259, 445, 282, 105, 12, 237, 110, 115, 388, 211, 468, 383, 492, 144, 594, 307, 366, 206, 126, 533, 324, 9, 463, 642

Offset: 2

M. F. Hasler, Nov 20 2016

nonn

It is conjectured that all integers >= 2 appear in A217727.

Cf. A217727, A114183.

A277848(n,a=2,u=[])={for(i=1,9e9,a==n&&return(i);u=setunion(u,[a]); while(#u>1&&u[2]==u[1]+1, u=u[^1]); my(t=log(a)\1); a=if(t>u[1]&&!setsearch(u, t), t, a^2))} \\ This code computes ("for each n") the sequence A217727 up to the index where n occurs.

A277848_vec(N,a=2,u=[],v=vector(N))={my(i=1,t); until(u[1]>N,a<=N&&v[a]=i; u=setunion(u,[a]); while(#u>1&&u[2]==u[1]+1, u=u[^1]);i++; a=if((t=log(a)\1)>u[1]&&!setsearch(u, t), t, a^2));v[^1]} \\ This is more efficient to compute a whole range a(1..N).

A213218 Even numbers in A221715.

Original entry on oeis.org

20, 192, 352, 104, 736, 264, 64, 464, 1088, 640, 1344, 800, 3136, 896, 1952, 2496, 22272, 2752, 1728, 1504, 8576, 4480, 3520, 12672, 7552, 4160, 3840, 11520, 2304, 19712, 9088, 8000, 10880, 14592, 11904, 4864, 21248, 8448, 17664, 26624, 10112, 6528, 5696, 6208, 24704, 22912, 28480, 9536, 39168, 41216, 36736

Offset: 1

N. J. A. Sloane, Mar 02 2013

nonn

Cf. A114183, A221715.

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

A268962 a(1) = 1 and a(2) = 2; thereafter a(n+1) = floor[sqrt(a(n))] if not already in the sequence; otherwise a(n+1) = a(n) + a(n-1).

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 21, 4, 25, 29, 54, 7, 61, 68, 129, 11, 140, 151, 12, 163, 175, 338, 18, 356, 374, 19, 393, 412, 20, 432, 452, 884, 1336, 36, 6, 42, 48, 90, 9, 99, 108, 10, 118, 128, 246, 15, 261, 16, 277, 293, 17, 310, 327, 637, 964, 31, 995, 1026, 32

Offset: 1

Melvin Peralta, Feb 16 2016

nonn

Hans Havermann found that the sequence is not a permutation of the integers since a(708)=a(1276)=1666.

Melvin Peralta, Table of n, a(n) for n = 1..20000

Cf. A014682, A114183, A000045

seq = {1, 2}; max; n = 3; While[n < max, If[MemberQ[seq, Floor[Sqrt[Last[seq]]]] == False, AppendTo[seq, Floor[Sqrt[Last[seq]]]], AppendTo[seq, Last[seq] + Part[seq, n - 2]]]; n++]

A268962_vec=A268962_used=[1,2]; A268962(n)={ #A268962_vec>=n || A268962_vec=concat(A268962_vec,vector(n-#A268962_vec)); A268962_vec[n] || for(n=#A268962_used+A268962_used[1],n, A268962_vec[n]&&next; (A268962_vec[n-1]<(A268962_used[1]+1)^2 || setsearch(A268962_used,A268962_vec[n]=sqrtint(A268962_vec[n-1]))) && A268962_vec[n]=A268962_vec[n-1]+A268962_vec[n-2]; A268962_used=setunion(A268962_used,A268962_vec[n..n]); while(#A268962_used>1 && A268962_used[2]==A268962_used[1]+1,A268962_used=A268962_used[^1])); A268962_vec[n]} \\ M. F. Hasler, Feb 16 2016

A335836 a(1) = 1; for n>1, a(n) = floor(a(n-1)^(1/3)) if that number is not already in the sequence, otherwise a(n) = 2*a(n-1).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 3, 6, 12, 24, 48, 96, 192, 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 13, 26, 52, 104, 208, 416, 7, 14, 28, 56, 112, 224, 448, 896, 9, 18, 36, 72, 144, 288, 576, 1152, 2304, 4608, 9216, 18432, 36864, 33, 66, 132, 264, 528, 1056, 2112

Offset: 1

Jinyuan Wang, Jun 27 2020

nonn

If k is not in this sequence, then none of k^(3^t), k^(3^t)+1, ..., (k+1)^(3^t)-1 belong to the sequence. Because (k+1)^(3^k) > 2*k^(3^k), any m > k^(3^k) is not in the sequence, which is a contradiction to {a(n)} is not bounded above. Therefore, this sequence is a permutation of the natural numbers.

Cf. A048766, A114183, A213912.

Nest[Append[#1, If[FreeQ[#1, #2], #2, 2 #1[[-1]] ]] & @@ {#, Floor[#[[-1]]^(1/3)]} &, {1}, 56] (* Michael De Vlieger, Jun 28 2020 *)

lista(nn) = {my(k, v=vector(nn)); v[1]=1; for(n=2, nn, if(vecsearch(vecsort(v), k=sqrtnint(v[n-1], 3)), v[n]=2*v[n-1], v[n]=k)); v; }

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

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

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Magma

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A213912 a(1) = 1; for n>1, a(n) = floor(sqrt(a(n-1))) if that number is not already in the sequence, otherwise a(n) = 3*a(n-1).

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Haskell

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A217727 a(1)=2. For n >= 1, let k = floor(log a(n)). If k >= 2 and k is not already in the sequence then a(n+1)=k, otherwise a(n+1)=a(n)^2.

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Maple

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A277848 Index at which n occurs in A217727, or 0 if there is no such index.

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PARI

PARI

A213218 Even numbers in A221715.

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

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Magma

Mathematica

PARI

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

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Author

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Crossrefs

Programs

Magma

Mathematica

PARI

A268962 a(1) = 1 and a(2) = 2; thereafter a(n+1) = floor[sqrt(a(n))] if not already in the sequence; otherwise a(n+1) = a(n) + a(n-1).

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Programs

Mathematica

PARI

A335836 a(1) = 1; for n>1, a(n) = floor(a(n-1)^(1/3)) if that number is not already in the sequence, otherwise a(n) = 2*a(n-1).

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