A036563 -id:A036563 - cp's OEIS frontend

A062001 Table by antidiagonals of n-Stohr sequences: T(n,k) is least positive integer not the sum of at most n distinct terms in the n-th row from T(n,1) through to T(n,k-1).

1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 5, 7, 4, 2, 1, 6, 10, 8, 4, 2, 1, 7, 13, 15, 8, 4, 2, 1, 8, 16, 22, 16, 8, 4, 2, 1, 9, 19, 29, 31, 16, 8, 4, 2, 1, 10, 22, 36, 46, 32, 16, 8, 4, 2, 1, 11, 25, 43, 61, 63, 32, 16, 8, 4, 2, 1, 12, 28, 50, 76, 94, 64, 32, 16, 8, 4, 2, 1, 13, 31, 57, 91, 125, 127, 64, 32, 16, 8, 4, 2, 1

Offset: 1

Henry Bottomley, May 29 2001

			Array begins as:
  1, 2, 3, 4,  5,  6,  7,   8,   9, ... A000027;
  1, 2, 4, 7, 10, 13, 16,  19,  22, ... A033627;
  1, 2, 4, 8, 15, 22, 29,  36,  43, ... A026474;
  1, 2, 4, 8, 16, 31, 46,  61,  76, ... A051039;
  1, 2, 4, 8, 16, 32, 63,  94, 125, ... A051040;
  1, 2, 4, 8, 16, 32, 64, 127, 190, ... ;
  1, 2, 4, 8, 16, 32, 64, 128, 255, ... ;
  1, 2, 4, 8, 16, 32, 64, 128, 256, ... ;
  1, 2, 4, 8, 16, 32, 64, 128, 256, ... ;
Antidiagonal triangle begins as:
   1;
   2,  1;
   3,  2,  1;
   4,  4,  2,  1;
   5,  7,  4,  2,   1;
   6, 10,  8,  4,   2,   1;
   7, 13, 15,  8,   4,   2,  1;
   8, 16, 22, 16,   8,   4,  2,  1;
   9, 19, 29, 31,  16,   8,  4,  2,  1;
  10, 22, 36, 46,  32,  16,  8,  4,  2, 1;
  11, 25, 43, 61,  63,  32, 16,  8,  4, 2, 1;
  12, 28, 50, 76,  94,  64, 32, 16,  8, 4, 2, 1;
  13, 31, 57, 91, 125, 127, 64, 32, 16, 8, 4, 2, 1;

G. C. Greubel, Antidiagonals n = 1..50, flattened

Rows include A000027, A033627, A026474, A051039, A051040.
Diagonals include A000079, A000225, A033484, A036563, A048487.
Cf. A048488, A048489, A048490, A048491, A139634, A139635, A139697.
A048483 can be seen as half this table.

T[n_, k_]:= If[kG. C. Greubel, May 03 2022 *)

def A062001(n,k):
    if (kA062001(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, May 03 2022

If k <= n+1 then A(n, k) = 2^(k-1), while if k > n+1, A(n, k) = (2^n - 1)*(k - n) + 1 (array).
T(n, k) = A(k, n-k+1) (antidiagonals).
T(2*n-1, n) = A000079(n-1), n >= 1.
T(2*n, n) = A000079(n), n >= 1.
T(2*n+1, n) = A000225(n+1), n >= 1.
T(2*n+2, n) = A033484(n), n >= 1.
T(2*n+3, n) = A036563(n+3), n >= 1.
T(2*n+4, n) = A048487(n), n >= 1.
From G. C. Greubel, May 03 2022: (Start)
T(n, k) = (2^k - 1)*(n-2*k+1) + 1 for k < n/2, otherwise 2^(n-k).
T(2*n+5, n) = A048488(n), n >= 1.
T(2*n+6, n) = A048489(n), n >= 1.
T(2*n+7, n) = A048490(n), n >= 1.
T(2*n+8, n) = A048491(n), n >= 1.
T(2*n+9, n) = A139634(n), n >= 1.
T(2*n+10, n) = A139635(n), n >= 1.
T(2*n+11, n) = A139697(n), n >= 1. (End)

A082693 Pyramidal sequence built with powers of 2.

Original entry on oeis.org

1, 2, 1, 2, 4, 2, 1, 2, 4, 8, 4, 2, 1, 2, 4, 8, 16, 8, 4, 2, 1, 2, 4, 8, 16, 32, 16, 8, 4, 2, 1, 2, 4, 8, 16, 32, 64, 32, 16, 8, 4, 2, 1, 2, 4, 8, 16, 32, 64, 128, 64, 32, 16, 8, 4, 2, 1, 2, 4, 8, 16, 32, 64, 128, 256, 128, 64, 32, 16, 8, 4, 2, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 256, 128

Offset: 1

Benoit Cloitre, Apr 12 2003

			Triangle begins:
                                 1
                               2 1 2
                             4 2 1 2 4
                           8 4 2 1 2 4 8
                        16 8 4 2 1 2 4 8 16
                     32 16 8 4 2 1 2 4 8 16 32
                  64 32 16 8 4 2 1 2 4 8 16 32 64
              128 64 32 16 8 4 2 1 2 4 8 16 32 64 128
          256 128 64 32 16 8 4 2 1 2 4 8 16 32 64 128 256
      512 256 128 64 32 16 8 4 2 1 2 4 8 16 32 64 128 256 512
1024 512 256 128 64 32 16 8 4 2 1 2 4 8 16 32 64 128 256 512 1024
... - _Philippe Deléham_, Mar 20 2013

Cf. A004738, A082693 (partial sums), A036563 (row sums).

pow2Pyram[row_] := Module[{st = 2^Range[0, row]}, Join[st, Reverse[Most[Rest[st]]]]]; Flatten[Array[pow2Pyram, 10]] (* Harvey P. Dale, May 09 2012 *)
Flatten[Table[Table[2^Abs[col], {col, -row, row}], {row, 0, 7}]] (* Alonso del Arte, Apr 15 2017 *)

for(i=0,9,forstep(j=i,0,-1,print1(1<Charles R Greathouse IV, Mar 20 2013

A109363 a(n) = 42^n - 3n - 5.

Original entry on oeis.org

-1, 0, 5, 18, 47, 108, 233, 486, 995, 2016, 4061, 8154, 16343, 32724, 65489, 131022, 262091, 524232, 1048517, 2097090, 4194239, 8388540, 16777145, 33554358, 67108787, 134217648, 268435373, 536870826, 1073741735, 2147483556, 4294967201, 8589934494, 17179869083, 34359738264

Offset: 0

Creighton Dement, Aug 22 2005

This sequence appears alongside the Eulerian numbers A000295 in the batch of sequences generated by the floretion given in the program code.

Floretion Algebra Multiplication Program, FAMP Code: 4ibaseisumseq[ - .5'i - .75'j - .5i' - .75j' + .25'ii' + .25'jj' - 1.25'kk' - .25'ik' + .5'jk' - .25'ki' + .5'kj' + .75e]; sumtype: Y[8] = (int)Y[6] - (int)Y[7] + Y[8] + sum (internal program code).

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000295, A036563.

A109363 := proc(n)
    4*2^n-3*n-5 ;
end proc:
seq(A109363(n),n=0..10) ; # R. J. Mathar, Jun 18 2019

f[n_]:=4*2^n-3*n-5; f[Range[0,20]] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2011 *)
LinearRecurrence[{4,-5,2},{-1,0,5},20] (* Harvey P. Dale, Jun 13 2011 *)

G.f.: (1-4*x)/((2*x-1)*(x-1)^2).
a(0)=-1, a(n) = 2*a(n-1) + 3*n - 1. - Vincenzo Librandi, Jan 29 2011
a(0)=-1, a(1)=0, a(2)=5, a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Harvey P. Dale, Jun 13 2011
a(n) - a(n-1) = A036563(n+1). - R. J. Mathar, Jun 18 2019
E.g.f.: exp(x)*(4*exp(x) - 3*x - 5). - Elmo R. Oliveira, Mar 07 2025

A226660 Smallest positive integer k with a primitive cycle of n positive integers (n>1) under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

1, 5, 7, 5, 11, 17, 13, 97, 59, 19, 55, 233, 11, 73, 25, 29, 47, 215, 41, 103, 145, 31, 13, 119, 131, 5, 47, 53, 67, 17, 337, 125, 115, 485, 133, 127, 49, 119, 191, 293, 133, 23, 79, 103, 191, 167, 91, 409, 329, 217, 109, 449, 241, 361, 353, 1303, 239, 149, 73

Offset: 2

Geoffrey H. Morley, Jul 05 2013

nonn

A cycle is called primitive if its elements are not a common multiple of the elements of another cycle.
The 3x+k function T_k is defined by T_k(x) = x/2 if x is even, (3x+k)/2 if x is odd, where k is odd.
For primitive cycles, GCD(k,6)=1.
For n>1, T_k has a primitive cycle of length n which includes 1 when k = A036563(n) = 2^n-3. So a(n) <= 2^n-3.

Geoffrey H. Morley, Table of n, a(n) for n = 2..3908

Cf. A226609, A226661, A226676.

A304370 Number of function calls of the first kind required to compute ack(3,n), where ack denotes the Ackermann function.

Original entry on oeis.org

9, 58, 283, 1244, 5213, 21342, 86367, 347488, 1394017, 5584226, 22353251, 89445732, 357848421, 1431524710, 5726360935, 22905967976, 91624920425, 366501778794, 1466011309419, 5864053626220, 23456231282029, 93824958682478, 375299901838703, 1501199741572464

Offset: 0

Olivier Gérard, May 11 2018

The distinction between different kinds of recursive calls is based on a naive implementation of the Ackermann function in C.
int ack(int m, int n)
{
// Final result
....if (m==0) return n + 1;
.
// Recursive calls of the first kind:
....if (n==0) return ack(m - 1, 1);
.
// Recursive calls of the second kind:
....return ack(m - 1, ack(m, n - 1));
}

Index entries for linear recurrences with constant coefficients, signature (8,-21,22,-8).

Cf. A036563, A074877, A304371.

G.f.: (8*x^2-14*x+9)/((4*x-1)*(2*x-1)*(x-1)^2). - Alois P. Heinz, May 12 2018

A304371 Number of function calls of the second kind required to compute ack(3,n), where ack denotes the Ackermann function.

Original entry on oeis.org

5, 47, 257, 1187, 5093, 21095, 85865, 346475, 1391981, 5580143, 22345073, 89429363, 357815669, 1431459191, 5726229881, 22905705851, 91624396157, 366500730239, 1466009212289, 5864049431939, 23456222893445, 93824941905287, 375299868284297, 1501199674463627

Offset: 0

Olivier Gérard, May 11 2018

The distinction between different kinds of recursive calls is based on a naive implementation of the Ackermann function in C.
int ack(int m, int n)
{
// Final result
....if (m==0) return n + 1;
.
// Recursive calls of the first kind:
....if (n==0) return ack(m - 1, 1);
.
// Recursive calls of the second kind:
....return ack(m - 1, ack(m, n - 1));
}

Index entries for linear recurrences with constant coefficients, signature (8,-21,22,-8).

Cf. A036563, A074877, A304370.

LinearRecurrence[{8,-21,22,-8},{5,47,257,1187},30] (* Harvey P. Dale, Oct 22 2019 *)

A304370(n) + a(n) + 1 = A074877(n).

G.f.: (8*x^3-14*x^2+7*x+5)/((4*x-1)*(2*x-1)*(x-1)^2). - Alois P. Heinz, May 12 2018

A367559 Square array T(n, k) = 2^k - n, read by ascending antidiagonals.

Original entry on oeis.org

1, 0, 2, -1, 1, 4, -2, 0, 3, 8, -3, -1, 2, 7, 16, -4, -2, 1, 6, 15, 32, -5, -3, 0, 5, 14, 31, 64, -6, -4, -1, 4, 13, 30, 63, 128, -7, -5, -2, 3, 12, 29, 62, 127, 256, -8, -6, -3, 2, 11, 28, 61, 126, 255, 512, -9, -7, -4, 1, 10, 27, 60, 125, 254, 511, 1024

Offset: 0

Paul Curtz, Nov 22 2023

			This sequence as square array T(n, k):
  n\k  0    1    2    3    4    5    6    7    8    9    10.
  ---------------------------------------------------------.
  0 :  1    2    4    8   16   32   64  128  256  512  1024.
  1 :  0    1    3    7   15   31   63  127  255  511  1023.
  2 : -1    0    2    6   14   30   62  126  254  510  1022.
  3 : -2   -1    1    5   13   29   61  125  253  509  1021.
  4 : -3   -2    0    4   12   28   60  124  252  508  1020.
  5 : -4   -3   -1    3   11   27   59  123  251  507  1019.
  6 : -5   -4   -2    2   10   26   58  122  250  506  1018.
  7 : -6   -5   -3    1    9   25   57  121  249  505  1017.
  8 : -7   -6   -4    0    8   24   56  120  248  504  1016.
  9 : -8   -7   -5   -1    7   23   55  119  247  503  1015.
  10: -9   -8   -6   -2    6   22   54  118  246  502  1014.

Paolo Xausa, Table of n, a(n) for n = 0..11475 (antidiagonals 0..150, flattened).

Cf. A000079, A000225, A000325, A000295.
Cf. A000325, A000918, A084634, A036563.
Cf. A168616, A185346, A246168.
Cf. A002262, A025581.

Table[2^k-n+k,{n,0,10},{k,0,n}] (* Paolo Xausa, Nov 28 2023 *)

T(n, k) = 2^k-n \\ Thomas Scheuerle, Nov 23 2023

G.f. of row n: 1/(1-2*x) - n/(1-x).
E.g.f. of row n: exp(2*x) - n*exp(x).
T(0, k) = 2^k = A000079(k).
T(1, k) = 2^k - 1 = A000225(k).
T(2, k) = 2^k - 2 = A000918(k).
T(3, k) = 2^k - 3 = A036563(k).
T(5, k) = 2^k - 5 = A168616(k).
T(9, k) = 2^k - 9 = A185346(k).
T(10, k) = 2^k - 10 = A246168(k).
T(n, k) = 3*T(n, k-1) - 2*T(n, k-2) for k > 1.
T(n+1, k) = T(n, k) + 1.
T(n, n) = 2^n - n = A000325(n).
Sum_{k = 0..n} T(n - k, k) = A084634(n).
a(n) = 2^A002262(n) - A025581(n).
G.f.: (1 - 2*x - y + 3*x*y)/((1 - x)^2*(1 - y)*(1 - 2*y)). - Stefano Spezia, Nov 27 2023

A091270 Smallest number having in binary representation a prefix of length n that is also a suffix of its successor.

Original entry on oeis.org

0, 2, 5, 13, 29, 61, 125, 253, 509, 1021, 2045, 4093, 8189, 16381, 32765, 65533, 131069, 262141, 524285, 1048573, 2097149, 4194301, 8388605, 16777213, 33554429, 67108861, 134217725, 268435453, 536870909, 1073741821

Offset: 0

Reinhard Zumkeller, Dec 27 2003

A091269(a(n)) = n and A091269(m) < n for m

a(n) = 2^(n+1) - 3 for n>2, cf. A036563.

Index entries for sequences related to binary expansion of n
Eric Weisstein's World of Mathematics, Binary
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A007088.

A093810 Smallest prime factor of 2^n-3.

Original entry on oeis.org

1, 5, 13, 29, 61, 5, 11, 509, 1021, 5, 4093, 19, 16381, 5, 13, 53, 11, 5, 1048573, 773, 4194301, 5, 16777213, 479, 37, 5, 11, 536870909, 23, 5, 9241, 29, 5113, 5, 242819, 47189, 11, 5, 13, 23, 47, 5, 5927, 2087, 227, 5, 11, 19, 59, 5, 13, 2203, 36217, 5, 181

Offset: 2

Yasutoshi Kohmoto, May 11 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 2..626 (terms 2..200 from Vincenzo Librandi)

Cf. A020639, A036563, A050414, A050415, A093817.

PrimeFactors[n_] := Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]; Table[ PrimeFactors[2^n - 3][[1]], {n, 2, 60}] (* Robert G. Wilson v, May 24 2004 *)
FactorInteger[#][[1,1]]&/@(2^Range[2,60]-3) (* Harvey P. Dale, Aug 21 2016 *)

a(n) = A020639(A036563(n)). - Amiram Eldar, Sep 12 2022

More terms from Robert G. Wilson v, May 24 2004

A093817 Largest prime factor of 2^n-3.

Original entry on oeis.org

1, 5, 13, 29, 61, 5, 23, 509, 1021, 409, 4093, 431, 16381, 6553, 71, 2473, 23831, 97, 1048573, 2713, 4194301, 1677721, 16777213, 70051, 5197, 31033, 1877171, 536870909, 46684427, 22605091, 464773, 296204641, 3360037, 6871947673, 283007

Offset: 2

Yasutoshi Kohmoto, May 11 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 2..626 (terms 2..200 from Vincenzo Librandi)

Cf. A006530, A036563, A050414, A050415, A093810.

PrimeFactors[n_] := Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]; Table[ PrimeFactors[2^n - 3][[ -1]], {n, 2, 46}] (* Robert G. Wilson v, May 24 2004 *)
Table[FactorInteger[2^n-3][[-1,1]],{n,2,40}] (* Harvey P. Dale, Feb 01 2015 *)

a(n) = A006530(A036563(n)). - Amiram Eldar, Sep 12 2022

More terms from Robert G. Wilson v, May 24 2004

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cp's OEIS Frontend

A062001 Table by antidiagonals of n-Stohr sequences: T(n,k) is least positive integer not the sum of at most n distinct terms in the n-th row from T(n,1) through to T(n,k-1).

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SageMath

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A082693 Pyramidal sequence built with powers of 2.

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A109363 a(n) = 4*2^n - 3*n - 5.

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Maple

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A226660 Smallest positive integer k with a primitive cycle of n positive integers (n>1) under iteration by the Collatz-like 3x+k function.

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A304370 Number of function calls of the first kind required to compute ack(3,n), where ack denotes the Ackermann function.

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A304371 Number of function calls of the second kind required to compute ack(3,n), where ack denotes the Ackermann function.

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Mathematica

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A367559 Square array T(n, k) = 2^k - n, read by ascending antidiagonals.

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PARI

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A091270 Smallest number having in binary representation a prefix of length n that is also a suffix of its successor.

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A093810 Smallest prime factor of 2^n-3.

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A109363 a(n) = 42^n - 3n - 5.