A050052 -id:A050052 - cp's OEIS frontend

A050032 a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique value such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = 1 and a(3) = 3.

1, 1, 3, 4, 7, 8, 11, 18, 29, 30, 33, 40, 51, 80, 113, 164, 277, 278, 281, 288, 299, 328, 361, 412, 525, 802, 1083, 1382, 1743, 2268, 3351, 5094, 8445, 8446, 8449, 8456, 8467, 8496, 8529, 8580, 8693, 8970, 9251, 9550, 9911, 10436

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..8193

Similar sequences with different initial conditions are A050024 (1,1,1), A050028 (1,1,2), A050036 (1,1,4), A050040 (1,2,1), A050044 (1,2,2), A050048 (1,2,3), A050052 (1,2,4), A050056 (1,3,1), A050060 (1,3,2), A050064 (1,3,3), A050068 (1,3,4).

a := proc(n) option remember;
`if`(n < 4, [1, 1, 3][n], a(n - 1) + a(-2^ceil(log[2](n - 1)) + 2*n - 3)):
end proc:
seq(a(n), n = 1..60); # Petros Hadjicostas, Nov 14 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 1, 3}, Flatten@Table[2 k - 1, {n, 5}, {k, 2^n}]] (* Ivan Neretin, Sep 07 2015 *)

Name edited by Petros Hadjicostas, Nov 14 2019

A050036 a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = 1 and a(3) = 4.

Original entry on oeis.org

1, 1, 4, 5, 9, 10, 14, 23, 37, 38, 42, 51, 65, 102, 144, 209, 353, 354, 358, 367, 381, 418, 460, 525, 669, 1022, 1380, 1761, 2221, 2890, 4270, 6491, 10761, 10762, 10766, 10775, 10789, 10826, 10868, 10933, 11077, 11430, 11788

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..8193

Same as A050052 and A050068 except for second term.

Cf. similar sequences with different initial conditions: A050024 (1,1,1), A050028 (1,1,2), A050032 (1,1,3), A050040 (1,2,1), A050044 (1,2,2), A050048 (1,2,3), A050052 (1,2,4), A050056 (1,3,1), A050060 (1,3,2), A050064 (1,3,3), A050068 (1,3,4).

a := proc(n) option remember;
`if`(n < 4, [1, 1, 4][n], a(n - 1) + a(-2^ceil(log[2](n - 1)) + 2*n - 3)):
end proc:
seq(a(n), n = 1..60); # Petros Hadjicostas, Nov 14 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 1, 4}, Flatten@Table[2 k - 1, {n, 5}, {k, 2^n}]] (* Ivan Neretin, Sep 07 2015 *)

Name edited by Petros Hadjicostas, Nov 14 2019

A050040 a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 2, and a(3) = 1.

Original entry on oeis.org

1, 2, 1, 2, 3, 4, 5, 8, 13, 14, 15, 18, 23, 36, 51, 74, 125, 126, 127, 130, 135, 148, 163, 186, 237, 362, 489, 624, 787, 1024, 1513, 2300, 3813, 3814, 3815, 3818, 3823, 3836, 3851, 3874, 3925, 4050, 4177, 4312, 4475, 4712, 5201

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..8193

Cf. similar sequences with different initial conditions: A050024 (1,1,1), A050028 (1,1,2), A050032 (1,1,3), A050036 (1,1,4), A050044 (1,2,2), A050048 (1,2,3), A050052 (1,2,4), A050056 (1,3,1), A050060 (1,3,2), A050064 (1,3,3), A050068 (1,3,4).

a := proc(n) option remember;
`if`(n < 4, [1, 2, 1][n], a(n - 1) + a(-2^ceil(log[2](n - 1)) + 2*n - 3)):
end proc:
seq(a(n), n = 1..60); # Petros Hadjicostas, Nov 14 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 2, 1}, Flatten@Table[2 k - 1, {n, 5}, {k, 2^n}]] (* Ivan Neretin, Sep 07 2015 *)

Name edited by Petros Hadjicostas, Nov 14 2019

A050044 a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = a(3) = 2.

Original entry on oeis.org

1, 2, 2, 3, 5, 6, 8, 13, 21, 22, 24, 29, 37, 58, 82, 119, 201, 202, 204, 209, 217, 238, 262, 299, 381, 582, 786, 1003, 1265, 1646, 2432, 3697, 6129, 6130, 6132, 6137, 6145, 6166, 6190, 6227, 6309, 6510, 6714, 6931, 7193, 7574, 8360

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..8193

Cf. similar sequences with different initial conditions: A050024 (1,1,1), A050028 (1,1,2), A050032 (1,1,3), A050036 (1,1,4), A050040 (1,2,1), A050048 (1,2,3), A050052 (1,2,4), A050056 (1,3,1), A050060 (1,3,2), A050064 (1,3,3), A050068 (1,3,4).

a := proc(n) option remember;
`if`(n < 4, [1, 2, 2][n], a(n - 1) + a(-2^ceil(log[2](n - 1)) + 2*n - 3)):
end proc:
seq(a(n), n = 1..60); # Petros Hadjicostas, Nov 14 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 2, 2}, Flatten@Table[2 k - 1, {n, 5}, {k, 2^n}]] (* Ivan Neretin, Sep 07 2015 *)

Name edited by Petros Hadjicostas, Nov 14 2019

A050048 a(n) = a(n-1) + a(m) for n >= 3, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = 2.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 11, 18, 29, 30, 33, 40, 51, 80, 113, 164, 277, 278, 281, 288, 299, 328, 361, 412, 525, 802, 1083, 1382, 1743, 2268, 3351, 5094, 8445, 8446, 8449, 8456, 8467, 8496, 8529, 8580, 8693, 8970, 9251, 9550, 9911, 10436

Offset: 1

Clark Kimberling

nonn

In the Mathematica program below, the author of the program uses the initial conditions a(1) = 1, a(2) = 2, and a(3) = 3. This is not necessary. We get the same sequence using the initial conditions a(1) = 1 and a(2) = 2. - Petros Hadjicostas, Nov 14 2019

Ivan Neretin, Table of n, a(n) for n = 1..8193

Cf. similar sequences with different initial conditions: A050024 (1,1,1), A050028 (1,1,2), A050032 (1,1,3), A050036 (1,1,4), A050040 (1,2,1), A050044 (1,2,2), A050052 (1,2,4), A050056 (1,3,1), A050060 (1,3,2), A050064 (1,3,3), A050068 (1,3,4).

a := proc(n) option remember;
`if`(n < 3, [1, 2][n], a(n - 1) + a(-2^ceil(log[2](n - 1)) + 2*n - 3)):
end proc:
seq(a(n), n = 1..60); # Petros Hadjicostas, Nov 14 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 2, 3}, Flatten@Table[2 k - 1, {n, 5}, {k, 2^n}]] (* Ivan Neretin, Sep 07 2015 *)

Name edited by Petros Hadjicostas, Nov 14 2019

A050056 a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 3, and a(3) = 1.

Original entry on oeis.org

1, 3, 1, 2, 3, 4, 5, 8, 13, 14, 15, 18, 23, 36, 51, 74, 125, 126, 127, 130, 135, 148, 163, 186, 237, 362, 489, 624, 787, 1024, 1513, 2300, 3813, 3814, 3815, 3818, 3823, 3836, 3851, 3874, 3925, 4050, 4177, 4312, 4475, 4712, 5201

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..8193

Cf. similar sequences with different initial conditions: A050024 (1,1,1), A050028 (1,1,2), A050032 (1,1,3), A050036 (1,1,4), A050040 (1,2,1), A050044 (1,2,2), A050048 (1,2,3), A050052 (1,2,4), A050060 (1,3,2), A050064 (1,3,3), A050068 (1,3,4).

a := proc(n) option remember; `if`(n < 4, [1, 3, 1][n],
       a(n - 1) + a(-2^ceil(log[2](n - 1)) + 2*n - 3)):
     end proc:
seq(a(n), n = 1..40); # Petros Hadjicostas, Nov 15 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 3, 1}, Flatten@Table[2 k - 1, {n, 5}, {k, 2^n}]] (* Ivan Neretin, Sep 07 2015 *)

Name edited by Petros Hadjicostas, Nov 15 2019

A050060 a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 3, and a(3) = 2.

Original entry on oeis.org

1, 3, 2, 3, 5, 6, 8, 13, 21, 22, 24, 29, 37, 58, 82, 119, 201, 202, 204, 209, 217, 238, 262, 299, 381, 582, 786, 1003, 1265, 1646, 2432, 3697, 6129, 6130, 6132, 6137, 6145, 6166, 6190, 6227, 6309, 6510, 6714, 6931, 7193, 7574, 8360

Offset: 1

Clark Kimberling

nonn

Variant of A050028 and A050044. - R. J. Mathar, Oct 15 2008

Ivan Neretin, Table of n, a(n) for n = 1..8193

Cf. similar sequences with different initial conditions: A050024 (1,1,1), A050028 (1,1,2), A050032 (1,1,3), A050036 (1,1,4), A050040 (1,2,1), A050044 (1,2,2), A050048 (1,2,3), A050052 (1,2,4), A050056 (1,3,1), A050064 (1,3,3), A050068 (1,3,4).

a := proc(n) option remember;
`if`(n < 4, [1, 3, 2][n], a(n - 1) + a(-2^ceil(log[2](n - 1)) + 2*n - 3)):
end proc:
seq(a(n), n = 1..60); # Petros Hadjicostas, Nov 14 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 3, 2}, Flatten@Table[2 k - 1, {n, 5}, {k, 2^n}]] (* Ivan Neretin, Sep 07 2015 *)

Name edited by Petros Hadjicostas, Nov 14 2019

A050064 a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = a(3) = 3.

Original entry on oeis.org

1, 3, 3, 4, 7, 8, 11, 18, 29, 30, 33, 40, 51, 80, 113, 164, 277, 278, 281, 288, 299, 328, 361, 412, 525, 802, 1083, 1382, 1743, 2268, 3351, 5094, 8445, 8446, 8449, 8456, 8467, 8496, 8529, 8580, 8693, 8970, 9251, 9550, 9911, 10436

Offset: 1

Clark Kimberling

nonn

Variant of A050032 or A050048. - R. J. Mathar, Oct 15 2008

Ivan Neretin, Table of n, a(n) for n = 1..8193

Cf. similar sequences with different initial conditions: A050024 (1,1,1), A050028 (1,1,2), A050032 (1,1,3), A050036 (1,1,4), A050040 (1,2,1), A050044 (1,2,2), A050048 (1,2,3), A050052 (1,2,4), A050056 (1,3,1), A050060 (1,3,2), A050068 (1,3,4).

a := proc(n) option remember;
     `if`(n < 4, [1, 3, 3][n], a(n - 1) + a(-2^ceil(log[2](n - 1)) + 2*n - 3)):
     end proc:
seq(a(n), n = 1..40); # Petros Hadjicostas, Nov 18 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 3, 3}, Flatten@Table[2 k - 1, {n, 5}, {k, 2^n}]] (* Ivan Neretin, Sep 07 2015 *)

Name edited by Petros Hadjicostas, Nov 18 2019

A050068 a(n) = a(n-1) + a(m) for n >= 3, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = 3.

Original entry on oeis.org

1, 3, 4, 5, 9, 10, 14, 23, 37, 38, 42, 51, 65, 102, 144, 209, 353, 354, 358, 367, 381, 418, 460, 525, 669, 1022, 1380, 1761, 2221, 2890, 4270, 6491, 10761, 10762, 10766, 10775, 10789, 10826, 10868, 10933, 11077, 11430, 11788

Offset: 1

Clark Kimberling

nonn

The author of the Mathematica program below uses the initial conditions a(1) = 1, a(2) = 3, and a(3) = 4. This is not necessary. We get the same sequence by using the initial conditions a(1) = 1 and a(2) = 3. - Petros Hadjicostas, Nov 15 2019

Ivan Neretin, Table of n, a(n) for n = 1..8193

Same as A050036 and A050052 except for the second term.

Cf. similar sequences with different initial conditions: A050024 (1,1,1), A050028 (1,1,2), A050032 (1,1,3), A050036 (1,1,4), A050040 (1,2,1), A050044 (1,2,2), A050048 (1,2,3), A050052 (1,2,4), A050056 (1,3,1), A050060 (1,3,2), A050064 (1,3,3).

a := proc(n) option remember; `if`(n < 3, [1,3][n],
        a(n - 1) + a(-2^ceil(log[2](n - 1)) + 2*n - 3))
     end proc:
seq(a(n), n = 1..40); # Petros Hadjicostas, Nov 15 2019

Fold[Append[#1, #1[[-1]] + #1[[#2]]] &, {1, 3, 4}, Flatten@Table[2 k - 1, {n, 5}, {k, 2^n}]] (* Ivan Neretin, Sep 07 2015 *)

Name edited by Petros Hadjicostas, Nov 15 2019

A161187 Let S = A089237\{0} = union of primes and nonzero squares; sequence gives indices of squares.

Original entry on oeis.org

1, 4, 7, 10, 14, 17, 22, 26, 31, 35, 41, 46, 52, 58, 63, 70, 78, 84, 91, 98, 106, 114, 122, 129, 139, 148, 156, 165, 175, 184, 193, 204, 214, 225, 235, 246, 256, 266, 279, 291, 304, 316, 326, 339, 351, 365, 376, 390, 406, 417, 429, 445, 462, 475, 489, 501, 514

Offset: 1

Daniel Tisdale, Jun 06 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A089237. Complement of A161188.

Cf. A050052.

a161187 n = a161187_list !! (n-1)
a161187_list = tail $ findIndices ((== 1) . a010052) a089237_list
-- Reinhard Zumkeller, Dec 18 2012

[1] cat [#PrimesUpTo(n^2-1)+n: n in [2..100]]; // Vincenzo Librandi, Feb 18 2016

Table[PrimePi[n^2 - 1] + n, {n, 60}] (* Vincenzo Librandi, Feb 18 2016 *)

from sympy import primepi
def A161187(n): return n+primepi(n**2) # Chai Wah Wu, Oct 12 2024

A089237(a(n)+1) = A000290(n). - Reinhard Zumkeller, Dec 18 2012

Edited by N. J. A. Sloane, Jun 07 2009

Extended by Ray Chandler, May 06 2010

cp's OEIS Frontend

A050032 a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique value such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = 1 and a(3) = 3.

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A050036 a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = 1 and a(3) = 4.

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A050040 a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 2, and a(3) = 1.

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A050044 a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = a(3) = 2.

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A050048 a(n) = a(n-1) + a(m) for n >= 3, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = 2.

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Maple

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A050056 a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 3, and a(3) = 1.

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Maple

Mathematica

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A050060 a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 3, and a(3) = 2.

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Maple

Mathematica

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A050064 a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = a(3) = 3.

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