A254050 -id:A254050 - cp's OEIS frontend

A254049 Odd bisection of A048673: a(n) = A048673(2*n-1).

1, 3, 4, 6, 13, 7, 9, 18, 10, 12, 28, 15, 25, 63, 16, 19, 33, 39, 21, 43, 22, 24, 88, 27, 61, 48, 30, 46, 58, 31, 34, 138, 60, 36, 73, 37, 40, 123, 72, 42, 313, 45, 67, 78, 49, 94, 93, 81, 51, 163, 52, 54, 193, 55, 57, 103, 64, 102, 213, 105, 85, 108, 172, 66, 118, 69, 127, 438, 70, 75, 133, 111, 109, 303

Offset: 1

Antti Karttunen, Jan 24 2015

nonn

Shift the prime factorization of odd numbers one step towards larger primes, add one and divide by two.

			For n = 8, the eighth odd number is 2*8 - 1 = 15 = 3*5 = prime(2) * prime(3). By adding one to both prime indices, we get prime(3) * prime(4) = 5*7 = 35, and (35+1)/2 = 18, thus a(8) = 18. Here prime(n) = A000040(n).

Cf. A032766 (omitting the initial 0, the same sequence sorted into ascending order).
Also a permutation of A253888.
Cf. A000040, A003961, A048673, A249735, A249746, A254050, A254051, A254053.

a(n) = A048673(2*n-1) = (1+A003961(2*n-1)) / 2 = (1+A249735(n)) / 2.

a(n) = A032766(A249746(n)).

A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n(2floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 3, 2, 4, 8, 5, 6, 11, 23, 14, 7, 17, 32, 68, 41, 9, 20, 50, 95, 203, 122, 10, 26, 59, 149, 284, 608, 365, 12, 29, 77, 176, 446, 851, 1823, 1094, 13, 35, 86, 230, 527, 1337, 2552, 5468, 3281, 15, 38, 104, 257, 689, 1580, 4010, 7655, 16403, 9842, 16, 44, 113, 311, 770, 2066, 4739, 12029, 22964, 49208, 29525, 18, 47

Offset: 1

L. Edson Jeffery & Antti Karttunen, Jan 24 2015

This is transposed dispersion of (3n-1), starting from its complement A032766 as the first row of square array A(row,col). Please see the transposed array A191450 for references and background discussion about dispersions.

For any odd number x = A135765(row,col), the result after one combined Collatz step (3x+1)/2 -> x (A165355) is found in this array at A(row+1,col).

			The top left corner of the array:
   1,   3,   4,   6,   7,   9,  10,  12,   13,   15,   16,   18,   19,   21
   2,   8,  11,  17,  20,  26,  29,  35,   38,   44,   47,   53,   56,   62
   5,  23,  32,  50,  59,  77,  86, 104,  113,  131,  140,  158,  167,  185
  14,  68,  95, 149, 176, 230, 257, 311,  338,  392,  419,  473,  500,  554
  41, 203, 284, 446, 527, 689, 770, 932, 1013, 1175, 1256, 1418, 1499, 1661
...

Inverse: A254052.
Transpose: A191450.
Row 1: A032766.
Cf. A007051, A057198, A199109, A199113 (columns 1-4).
Cf. A254046 (row index of n in this array, see also A253786), A253887 (column index).
Array A135765(n,k) = 2*A(n,k) - 1.
Other related arrays: A254055, A254101, A254102.
Cf. also A000079, A000244, A003961, A007310, A032766, A002260, A004736, A038722, A165355, A254050.
Related permutations: A048673, A254053, A183209, A249745, A254103, A254104.

In A(n,k)-formulas below, n is the row, and k the column index, both starting from 1:
A(n,k) = (3 + ( A000244(n) * (2*A032766(k) - 1) )) / 6. - Antti Karttunen after L. Edson Jeffery's direct formula for A191450, Jan 24 2015
A(n,k) = A048673(A254053(n,k)). [Alternative formula.]
A(n,k) = (1/2) * (1 + A003961((2^(n-1)) * A254050(k))). [The above expands to this.]
A(n,k) = (1/2) * (1 + (A000244(n-1) * A007310(k))). [Which further reduces to this, equivalent to L. Edson Jeffery's original formula above.]
A(1,k) = A032766(k) and for n > 1: A(n,k) = (3 * A254051(n-1,k)) - 1. [The definition of transposed dispersion of (3n-1).]
A(n,k) = (1+A135765(n,k))/2, or when expressed one-dimensionally, a(n) = (1+A135765(n))/2.
A(n+1,k) = A165355(A135765(n,k)).
As a composition of related permutations. All sequences interpreted as one-dimensional:
a(n) = A048673(A254053(n)). [Proved above.]
a(n) = A191450(A038722(n)). [Transpose of array A191450.]

A254053 Square array: A(row,col) = 2^(row-1) * ((2*A249745(col))-1) = A064216(A254051(row,col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 3, 2, 5, 6, 4, 7, 10, 12, 8, 11, 14, 20, 24, 16, 13, 22, 28, 40, 48, 32, 17, 26, 44, 56, 80, 96, 64, 19, 34, 52, 88, 112, 160, 192, 128, 9, 38, 68, 104, 176, 224, 320, 384, 256, 23, 18, 76, 136, 208, 352, 448, 640, 768, 512, 29, 46, 36, 152, 272, 416, 704, 896, 1280, 1536, 1024, 15, 58, 92, 72, 304, 544, 832, 1408, 1792, 2560, 3072, 2048, 31, 30

Offset: 1

Antti Karttunen, Jan 24 2015

Shares with A135764 and A253551 the property that A001511(n) = k for all terms n on row k and when going downward in each column, terms grow by doubling.

			The top left corner of the array:
   1,  3,  5,   7,  11,  13,  17,  19,   9,  23,  29,  15,  31,  37,  41,  43,
   2,  6, 10,  14,  22,  26,  34,  38,  18,  46,  58,  30,  62,  74,  82,  86,
   4, 12, 20,  28,  44,  52,  68,  76,  36,  92, 116,  60, 124, 148, 164, 172,
   8, 24, 40,  56,  88, 104, 136, 152,  72, 184, 232, 120, 248, 296, 328, 344,
  16, 48, 80, 112, 176, 208, 272, 304, 144, 368, 464, 240, 496, 592, 656, 688,
...

Inverse: A254054.
Similar or related permutations: A135764, A253551, A064216, A254051.
Cf. A000079, A001511, A002260, A004736, A064989, A135765, A249745, A254049, A254050.

A(row,col) = A135764(row, A249745(col)). [Is otherwise the same array as A135764, but the column positions have been permuted by A249745.]
A(row,col) = 2^(row-1) * ((2*A249745(col))-1) = 2^(row-1) * A254050(col). [The above expands to this.]
a(n) = A064989(A135765(n)).
As a composition of other permutations:
a(n) = A064216(A254051(n)). [As an array: A(row,col) = A064216(A254051(row,col)).]

A273664 a(n) = A249746(A032766(n)).

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 10, 17, 11, 13, 26, 14, 15, 16, 18, 41, 20, 31, 21, 23, 40, 24, 25, 27, 48, 28, 30, 45, 33, 63, 54, 34, 35, 36, 37, 38, 43, 68, 70, 57, 115, 44, 46, 85, 47, 50, 74, 73, 51, 53, 87, 55, 107, 56, 58, 97, 60, 180, 61, 64, 96, 83, 65, 66, 67, 71, 114, 101, 100, 75, 110, 136, 108, 76, 77, 78, 80, 124, 81

Offset: 1

Antti Karttunen, Aug 06 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A032766, A249745, A249746, A249824, A254049, A254050, A275716.

Cf. also A273669 (natural numbers not in this sequence).

t = PositionIndex[FactorInteger[#][[1, 1]] & /@ Range[10^6]]; f[n_] := Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; Flatten@ Map[Position[Lookup[t, FactorInteger[#][[1, 1]]], #] &[f@ f[2 #]] &, Map[Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@#, Last@#] &@ Transpose@ FactorInteger[2 # - 1] &, Floor[#/2] + # & /@ Range@ 80]] (* Michael De Vlieger, Aug 07 2016, Version 10 *)

(define (A273664 n) (A249746 (A032766 n)))

a(n) = A249746(A032766(n)).
a(n) = A249824(A254050(n)).
a(n) = A249746(A254049(A249745(n))).

cp's OEIS Frontend

A254049 Odd bisection of A048673: a(n) = A048673(2*n-1).

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A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n(2floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A254053 Square array: A(row,col) = 2^(row-1) * ((2*A249745(col))-1) = A064216(A254051(row,col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A273664 a(n) = A249746(A032766(n)).

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cp's OEIS Frontend

A254049 Odd bisection of A048673: a(n) = A048673(2*n-1).

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Formula

A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n*(2*floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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Formula

A254053 Square array: A(row,col) = 2^(row-1) * ((2*A249745(col))-1) = A064216(A254051(row,col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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Formula

A273664 a(n) = A249746(A032766(n)).

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Author

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Programs

Mathematica

Scheme

Formula

A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n(2floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...