A153893 -id:A153893 - cp's OEIS frontend

A182465 a(n) = 3a(n-1) - 2a(n-2) with a(0)=28 and a(1)=70.

28, 70, 154, 322, 658, 1330, 2674, 5362, 10738, 21490, 42994, 86002, 172018, 344050, 688114, 1376242, 2752498, 5505010, 11010034, 22020082, 44040178, 88080370, 176160754, 352321522, 704643058, 1409286130, 2818572274, 5637144562, 11274289138, 22548578290

Offset: 0

Odimar Fabeny, Apr 30 2012

Number of vertices into building blocks of 3d objects with 7 vertices.

			a(0) = 7+14+7;
a(0) = 7+14+28+14+7;
a(0) = 7+14+28+56+28+14+7;
a(0) = 7+14+28+56+112+56+28+14+7.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000045, A028399, A038578, A089143, A173033, A182461, A182462, A182464, A182466, A182467.

CoefficientList[Series[-((14 (x - 2))/(2 x^2 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 01 2014 *)
LinearRecurrence[{3,-2},{28,70},30] (* Harvey P. Dale, Oct 05 2015 *)

a(n) = a(n-1)*2 + 14.
a(n) = 14*A153893(n). - Michel Marcus, Jun 01 2014
G.f.: -((14*(x-2))/(2*x^2-3*x+1)). - Vincenzo Librandi, Jun 01 2014

A283508 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 10, 101, 1011, 10111, 101111, 1011111, 10111111, 101111111, 1011111111, 10111111111, 101111111111, 1011111111111, 10111111111111, 101111111111111, 1011111111111111, 10111111111111111, 101111111111111111, 1011111111111111111, 10111111111111111111

Offset: 0

Robert Price, Mar 09 2017

Initialized with a single black (ON) cell at stage zero.

As far as the b-files reach (125 terms) this is the same as A267623. - R. J. Mathar, Mar 17 2017

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A267623, A036563, A153893.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 643; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

Conjectures from Colin Barker, Mar 10 2017: (Start)
G.f.: (1 - x + x^2) / ((1 - x)*(1 - 10*x)).
a(n) = (91*10^n - 10) / 90 for n>0.
a(n) = 11*a(n-1) - 10*a(n-2) for n>2.
(End)
Equivalent conjecture: a(n) = A267623(n). - R. J. Mathar, Mar 17 2017

A365802 Numbers k such that A163511(k) is a fifth power.

Original entry on oeis.org

0, 16, 33, 67, 135, 271, 512, 543, 1025, 1056, 1087, 2051, 2113, 2144, 2175, 4103, 4227, 4289, 4320, 4351, 8207, 8455, 8579, 8641, 8672, 8703, 16384, 16415, 16911, 17159, 17283, 17345, 17376, 17407, 32769, 32800, 32831, 33792, 33823, 34319, 34567, 34691, 34753, 34784, 34815, 65539, 65601, 65632, 65663, 67585, 67616

Offset: 1

Antti Karttunen, Oct 01 2023

nonn

Equivalently, numbers k for which A332214(k), and also A332817(k) are fifth powers.
The sequence is defined inductively as:
 (a) it contains 0 and 16, and
 (b) for any nonzero term a(n), (2*a(n)) + 1 and 32*a(n) are also included as terms.
When iterating n -> 2n+1 mod 31, starting from 16 we obtain five distinct remainders 16, 2, 5, 11, 23, before the cycle starts again from 16. (see A153893), while x^5 mod 31 may obtain only these values: 0, 1, 5, 6, 25, 26, 30. The only common element of these sets is 5. We have x^5 == 5 (mod 31) whenever x == 7, 14, 19, 25, 28 mod 31, with all other x leaving a remainder that is not in the set [16, 2, 5, 11, 23].
On the other hand, when iterating n -> 2n+1 mod 33, starting from 16 we obtain ten distinct remainders 16, 0, 1, 3, 7, 15, 31, 30, 28, 24, before the cycle starts again from 16, while x^5 mod 33 obtain only these values: 0, 1, 10, 11, 12, 21, 22, 23, 32. We have x^5 == 0 (mod 33) iff x == 0 (mod 33) and x^5 == 1 (mod 33) whenever x == 1, 4, 16, 25, 31 mod 33. In the n->2n+1 cycles of 5 and 10 elements starting from 16, the 5's (of every second cycle) in the former and the 1's in the latter are aligned with each other.
In any case, this sequence do not contain any fifth powers after the initial zero. See A365805. - Antti Karttunen, Nov 23 2023

Index entries for sequences related to binary expansion of n

Positions of multiples of 5 in A365805.
Sequence A243071(n^5), n >= 1, sorted into ascending order.
Subsequences: A013825, A198275.
Cf. A000584, A153893, A163511, A332214, A332817.
Cf. also A365801, A365808, A366287, A366391.

A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
isA365802(n) = ispower(A163511(n),5);

isA365802(n) = if(n<=16, !(n%16), if(n%2, isA365802((n-1)/2), if(n%32, 0, isA365802(n/32))));

A020944 a(2n+1) = |a(2n) - a(2n-1)|, a(2n) = a(n) + a(n-1), a(0) = -1.

Original entry on oeis.org

-1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3

Offset: 0

Clark Kimberling

a(n) = abs(t(n+1)) if n>0 where t(n) is the twisted Stern sequence defined by R. Bacher and M. Coons. - Michael Somos, Jan 08 2011

a(A153893(n)) = 0. - Reinhard Zumkeller, Mar 13 2011

			G.f. = -1 + x + x^3 + x^4 + x^6 + x^7 + 2*x^8 + x^9 + x^10 + x^12 + x^13 + 2*x^14 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..12287
Roland Bacher, Twisting the Stern sequence, arXiv:1005.5627v1 [math.CO], 2010.
Michael Coons, On Some Conjectures concerning Stern's Sequence and its Twist, arXiv:1008.0193v3 [math.NT], 2010.

Cf. A020947, A020948, A020949, A020950, A153893.

a020944 n = a020944_list !! n
a020944_list = -1 : f [1,0] where f (x:y:xs) = x : f (y:xs ++ [x,x+y])
-- Same list generator function as for a020951_list, cf. A020951.
-- Reinhard Zumkeller, Mar 13 2013

a[ n_] := Which[ n < 2, Boole[n == 1] - Boole[n == 0], OddQ[n], Abs[a[n - 1] - a[n - 2]], True, a[n/2] + a[n/2 - 1]]; (* Michael Somos, Jul 25 2018 *)

{a(n) = if( n<2,(n==1) - (n==0),  n%2, abs( a(n-1) - a(n-2) ), a(n/2) + a(n/2 - 1) )}; /* Michael Somos, Jan 08 2011 */

{a(n) = my(A, m); if( n<0, 0, m = 1; A = -1 + O(x); while( m <= n, m*=2; A = 2*x + (1 + x + x^2) * subst( A, x, x^2 ) ); polcoeff( A, n ) )}; /* Michael Somos, Jan 08 2011 */

G.f. A(x) satisfies: A(x) = 2*x + (1 + x + x^2) * A(x^2). - Michael Somos, Jan 08 2011

More terms from Henry Bottomley, May 16 2001

Added a(0) from Michael Somos, Jan 08 2011

A290114 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943, 12884901887

Offset: 0

Robert Price, Jul 19 2017

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for linear recurrences with constant coefficients, signature (3,-2).
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Essentially the same as A153893, A083329, A055010, A052940, A266550.

Cf. A290111, A290112, A290113, .

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 643; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

For n>1, a(n) = 3*2^(n-1)-1.
a(n) = A266550(n+2) for n > 1. - Georg Fischer, Oct 30 2018
a(n) = 2*a(n-1) + 1 for n=1 and n>=3. - Gennady Eremin, Aug 26 2023
From Chai Wah Wu, Apr 02 2024: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n > 3.
G.f.: (2*x^3 - 2*x^2 + 1)/((x - 1)*(2*x - 1)). (End)

A368085 Square array read by ascending antidiagonals: row n is the trajectory of P under the 'Px+1' map, where P = n-th prime.

Original entry on oeis.org

2, 3, 5, 5, 10, 11, 7, 26, 5, 23, 11, 50, 13, 16, 47, 13, 122, 25, 66, 8, 95, 17, 170, 61, 5, 33, 4, 191, 19, 290, 85, 672, 1, 11, 2, 383, 23, 362, 145, 17, 336, 8, 56, 1, 767, 29, 530, 181, 29, 222, 168, 4, 28, 4, 1535, 31, 842, 265, 3440, 494, 111, 84, 2, 14, 2, 3071

Offset: 1

Paolo Xausa, Dec 11 2023

The 'Px+1 map' is defined as follows: if there exists p = smallest prime < P which divides x then x = x/p, otherwise x = P*x + 1.

			Array begins:
  [ 1]   2,   5,  11,    23,   47,   95, 191, 383,  767, ... = A153893
  [ 2]   3,  10,   5,    16,    8,    4,   2,   1,    4, ... = A033478
  [ 3]   5,  26,  13,    66,   33,   11,  56,  28,   14, ... = A057688
  [ 4]   7,  50,  25,     5,    1,    8,   4,   2,    1, ... = A368113
  [ 5]  11, 122,  61,   672,  336,  168,  84,  42,   21, ... = A368114
  [ 6]  13, 170,  85,    17,  222,  111,  37, 482,  241, ... = A057684
  [ 7]  17, 290, 145,    29,  494,  247,  19, 324,  162, ... = A368115
  [ 8]  19, 362, 181,  3440, 1720,  860, 430, 215,   43, ... = A057685
  [ 9]  23, 530, 265,    53, 1220,  610, 305,  61, 1404, ... = A057686
  [10]  29, 842, 421, 12210, 6105, 2035, 407,  37, 1074, ... = A057687
  ...    |    |    |
      A000040 | A066885 (from n = 2)
           A066872

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150, flattened).

Rows 1-10: A153893, A033478, A057688, A368113, A368114, A057684, A368115, A057685, A057686, A057687.
Columns 1-3: A000040, A066872, A066885 (from n = 2).
Main diagonal gives A368159.
Cf. A057689, A057690, A057691.

Px1[p_,n_]:=Catch[For[i=1,iA368085list[dmax_]:=With[{a=Reverse[Table[NestList[Px1[Prime[n],#]&,Prime[n],dmax-n],{n,dmax}]]},Array[Diagonal[a,#]&,dmax,1-dmax]];
A368085list[15] (* Generates 15 antidiagonals *)

A366287 Numbers k such that A163511(k) is a seventh power.

Original entry on oeis.org

0, 64, 129, 259, 519, 1039, 2079, 4159, 8192, 8319, 16385, 16512, 16639, 32771, 33025, 33152, 33279, 65543, 66051, 66305, 66432, 66559, 131087, 132103, 132611, 132865, 132992, 133119, 262175, 264207, 265223, 265731, 265985, 266112, 266239, 524351, 528415, 530447, 531463, 531971, 532225, 532352, 532479, 1048576, 1048703

Offset: 1

Antti Karttunen, Oct 09 2023

nonn

Equivalently, numbers k for which A332214(k), and also A332817(k) are seventh powers.
The sequence is defined inductively as:
 (a) it contains 0 and 64,
   and
 (b) for any nonzero term a(n), (2*a(n)) + 1 and 128*a(n) are also included as terms.
When iterating n -> 2n+1 mod 127, starting from 64 we get 64, 2, 5, 11, 23, 47, 95, and then cycle starts again from 64 (see A153893), while on the other hand, x^7 mod 127 obtains values: 0, 1, 19, 20, 22, 24, 28, 37, 52, 59, 68, 75, 90, 99, 103, 105, 107, 108, 126. These sets have no terms in common, therefore there are no seventh powers in this sequence after the initial 0.

Positions of multiples of 7 in A365805.
Sequence A243071(n^7), n >= 1, sorted into ascending order.
Cf. A001015, A153893, A163511, A332214, A332817.
Cf. A365801, A365802, A365808, A366391.

A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
isA366287(n) = ispower(A163511(n),7);

isA366287(n) = if(n<=64, !(n%64), if(n%2, isA366287((n-1)/2), if(n%128, 0, isA366287(n>>7))));

A366391 Numbers k such that A163511(k) is an eleventh power.

Original entry on oeis.org

0, 1024, 2049, 4099, 8199, 16399, 32799, 65599, 131199, 262399, 524799, 1049599, 2097152, 2099199, 4194305, 4196352, 4198399, 8388611, 8392705, 8394752, 8396799, 16777223, 16785411, 16789505, 16791552, 16793599, 33554447, 33570823, 33579011, 33583105, 33585152, 33587199, 67108895, 67141647, 67158023, 67166211, 67170305

Offset: 1

Antti Karttunen, Oct 09 2023

nonn

Equivalently, numbers k for which A332214(k), and also A332817(k) are eleventh powers.
The sequence is defined inductively as:
 (a) it contains 0 and 1024,
   and
 (b) for any nonzero term a(n), (2*a(n)) + 1 and 2048*a(n) are also included as terms.
When iterating n -> 2n+1 mod 2047, starting from 1024 we get 1024, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535,  and then cycle starts again from 1024 (see A153893), while on the other hand, x^11 mod 2047 obtains values: 0, 1, 230, 322, 344, 368, 390, 482, 622, 712, 713, 942, 967, 1013, 1034, 1080, 1105, 1334, 1335, 1425, 1565, 1657, 1679, 1703, 1725, 1817, 2046. These sets have no terms in common, therefore there are no eleventh powers in this sequence after the initial 0.

Positions of multiples of 11 in A365805.
Sequence A243071(n^11), n >= 1, sorted into ascending order.
Cf. A008455, A153893, A163511, A332214, A332817.
Cf. also A365801, A365802, A365808, A366287.

A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
isA366391v(n) = ispower(A163511(n),11);

isA366391(n) = if(n<=1024, !(n%1024), if(n%2, isA366391((n-1)/2), if(n%2048, 0, isA366391(n>>11))));

A139244 a(0) = 4; a(n) = a(n-1)^2 - 1.

Original entry on oeis.org

4, 15, 224, 50175, 2517530624, 6337960442777829375, 40169742574216538983356186036612890624, 1613608218478824775913354216413699241125577233045500390244103887844987109375

Offset: 0

Jonathan Vos Post, Jun 06 2008

This is the next analog of A003096 with different initial value a(0), as starting with a(0) = 2 is A003096 and a(0) = 3 is A003096 with first term omitted. It alternates between even and odd values, specifically between 4 mod 10 and 5 mod 10 and is always composite (by difference of squares factorization).

a(n+2) is divisible by a(n)^2. A007814(a(2 n)) = A153893(n). - Robert Israel, Jul 20 2015

Robert Israel, Table of n, a(n) for n = 0..10
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A003096, A007814, A153893, A260315.

A[0]:= 4:
for n from 1 to 10 do A[n]:= A[n-1]^2-1 od:
seq(A[i],i=0..10); # Robert Israel, Jul 20 2015

a=4; lst={a}; Do[b=a^2-1; AppendTo[lst,b]; a=b, {n,10}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 28 2010 *)

a(n)=if(n,a(n-1)^2-1,4) \\ Charles R Greathouse IV, Jul 23 2015

a(n-1) = ceiling(c^(2^n)) where c is a constant between 1 and 2.

More specifically, c=1.9668917617901763653335057202... (sequence A260315). - Chayim Lowen, Jul 17 2015

A266550 Independence number of the n-Mycielski graph.

Original entry on oeis.org

1, 1, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943, 12884901887

Offset: 1

Eric W. Weisstein, Dec 31 2015

Eric Weisstein's World of Mathematics, Independence Number.
Eric Weisstein's World of Mathematics, Mycielski Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A052940, A055010, A083329, A153893.

[1,1] cat [-1+3*2^(n-3): n in [3..40]]; // Vincenzo Librandi, Jan 01 2016

I:=[1,1,2,5]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jan 01 2016

Table[Piecewise[{{-1 + 3 2^(n - 3), n > 2}}, 1], {n, 35}]
CoefficientList[Series[1 + x*(1 - x + x^2)/((1 - x)*(1 - 2*x)), {x, 0, 35}], x] (* Vincenzo Librandi, Jan 01 2016 *)

a(1) = 1, a(2) = 1; for n>2, a(n) = -1 + 3*2^(n-3) = A083329(n-2) = A055010(n-2) = A153893(n-3).
G.f.: x + x^2*(1 - x + x^2)/((1 - x)*(1 - 2*x)).
a(n) = 3*a(n-1)-2*a(n-2) for n>2. - Vincenzo Librandi, Jan 01 2016
a(n) = A052940(n-3) for n > 3. - Georg Fischer, Oct 23 2018
E.g.f.: (3*exp(2*x) - 8*exp(x) + 5 + 10*x+ 2*x^2)/8. - Stefano Spezia, Sep 14 2024

cp's OEIS Frontend

A182465 a(n) = 3a(n-1) - 2a(n-2) with a(0)=28 and a(1)=70.

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A283508 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.

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A365802 Numbers k such that A163511(k) is a fifth power.

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A020944 a(2n+1) = |a(2n) - a(2n-1)|, a(2n) = a(n) + a(n-1), a(0) = -1.

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A290114 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.

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A368085 Square array read by ascending antidiagonals: row n is the trajectory of P under the 'Px+1' map, where P = n-th prime.

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A366287 Numbers k such that A163511(k) is a seventh power.

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A366391 Numbers k such that A163511(k) is an eleventh power.

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