cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A135764 Distribute the natural numbers in columns based on the occurrence of "2" in each prime factorization; square array A(row,col) = 2^(row-1) * ((2*col)-1), read by descending antidiagonals.

Original entry on oeis.org

1, 3, 2, 5, 6, 4, 7, 10, 12, 8, 9, 14, 20, 24, 16, 11, 18, 28, 40, 48, 32, 13, 22, 36, 56, 80, 96, 64, 15, 26, 44, 72, 112, 160, 192, 128, 17, 30, 52, 88, 144, 224, 320, 384, 256, 19, 34, 60, 104, 176, 288, 448, 640, 768, 512, 21, 38, 68, 120, 208, 352, 576, 896, 1280, 1536, 1024, 23, 42, 76, 136, 240, 416, 704, 1152, 1792, 2560, 3072, 2048, 25, 46, 84, 152, 272, 480, 832, 1408, 2304, 3584, 5120, 6144, 4096, 27, 50, 92, 168, 304, 544, 960, 1664, 2816
Offset: 1

Views

Author

Alford Arnold, Nov 29 2007

Keywords

Comments

The array in A135764 is identical to the array in A054582 [up to the transposition and different indexing. - Clark Kimberling, Dec 03 2010; comment amended by Antti Karttunen, Feb 03 2015; please see the illustration in Example section].
The array gives a bijection between the natural numbers N and N^2. A more usual bijection is to take the natural numbers A000027 and write them in the usual OEIS square array format. However this bijection has the advantage that it can be formed by iterating the usual bijection between N and 2N. - Joshua Zucker, Nov 04 2011
The array can be used to determine the configurations of k-th Towers of Hanoi moves, by labeling odd row terms C,B,A,C,B,A,... and even row terms B,C,A,B,C,A,.... Then given k equal to or greater than term "a" in each n-th row, but less than the next row term, record the label A, B, or C for term "a". This denotes the peg position for the disc corresponding to the n-th row. For example, with k = 25, five discs are in motion since the binary for 25 = 11001, five bits. We find that 25 in row 5 is greater than 16 labeled C, but less than 48. Thus, disc 5 is on peg C. In the 4th row, 25 is greater than 24 (a C), but less than 40, so goes onto the C peg. Similarly, disc 3 is on A, 2 is on A, and disc 1 is on A. Thus, discs 2 and 3 are on peg A, while 1, 4, and 5 are on peg C. - Gary W. Adamson, Jun 22 2012
Shares with arrays A253551 and A254053 the property that A001511(n) = k for all terms n on row k and when going downward in each column, terms grow by doubling. - Antti Karttunen, Feb 03 2015
Let P be the infinite palindromic word having initial word 0 and midword sequence (1,2,3,4,...) = A000027. Row n of the array A135764 gives the positions of n-1 in S. ("Infinite palindromic word" is defined at A260390.) - Clark Kimberling, Aug 13 2015
The probability distribution series 1 = 2/3 + 4/15 + 16/255 + 256/65535 + ... + A001146(n-1)/A051179(n) governs the proportions of terms in A001511 from row n of the array. In A001511(1..15) there are ((2/3) * 15) = ten terms from row one of the array, ((4/15) * 15) = four terms from row two, and ((16/255) * 15) = one (rounded), giving one term from row three (a 4). - Gary W. Adamson, Dec 16 2021
From Gary W. Adamson, Dec 30 2021: (Start)
Subarrays representing the number of divisors of an integer can be mapped on the table. For 60, write the odd divisors on the top row: 1, 3, 5, 15. Since 60 has 12 divisors, let the left column equal 1, 2, 4, where 4 is the highest power of 2 dividing 60. Multiplying top row terms by left column terms, we get the result:
1 3 5 15
2 6 10 30
4 12 20 60. (End)

Examples

			The table begins
   1,  3,   5,   7,   9,  11,  13,  15,  17,  19,  21,  23, ...
   2,  6,  10,  14,  18,  22,  26,  30,  34,  38,  42,  46, ...
   4, 12,  20,  28,  36,  44,  52,  60,  68,  76,  84,  92, ...
   8, 24,  40,  56,  72,  88, 104, 120, 136, 152, 168, 184, ...
  16, 48,  80, 112, 144, 176, 208, 240, 272, 304, 336, 368, ...
  32, 96, 160, 224, 288, 352, 416, 480, 544, 608, 672, 736, ...
etc.
For n = 6, we have [A002260(6), A004736(6)] = [3, 1] (i.e., 6 corresponds to location 3,1 (row,col) in above table) and A(3,1) = A000079(3-1) * A005408(1-1) = 2^2 * 1 = 4.
For n = 13, we have [A002260(13), A004736(13)] = [3, 3] (13 corresponds to location 3,3 (row,col) in above table) and A(3,3) = A000079(3-1) * A005408(3-1) = 2^2 * 5 = 20.
For n = 23, we have [A002260(23), A004736(23)] = [2, 6] (23 corresponds to location 2,6) and A(2,6) = A000079(2-1) * A005408(6-1) = 2^1 * 11 = 22.

Links

Crossrefs

Transpose: A054582.
Inverse permutation: A249725.
Column 1: A000079.
Row 1: A005408.
Cf. A001511 (row index), A003602 (column index, both one-based).
Related arrays: A135765, A253551, A254053, A254055.
Cf. A000027, A002260, A004736, A064989, A135766, A249746.
Cf. also permutations A246675, A246676, A249741, A249811, A249812.
Cf. A260390.
Cf. A001146, A051179.

Programs

  • Maple
    seq(seq(2^(j-1)*(2*(i-j)+1),j=1..i),i=1..20); # Robert Israel, Feb 03 2015
  • Mathematica
    f[n_] := Block[{i, j}, {1}~Join~Flatten@ Last@ Reap@ For[j = 1, j <= n, For[i = j, i > 0, Sow[2^(j - i - 1)*(2 i + 1)], i--], j++]]; f@ 10 (* Michael De Vlieger, Feb 03 2015 *)
  • PARI
    a(n) = {s = ceil((1 + sqrt(1 + 8*n)) / 2); r = n - binomial(s-1, 2) - 1;k = s - r - 2; 2^r * (2 * k + 1) } \\ David A. Corneth, Feb 05 2015
  • Scheme
    (define (A135764 n) (A135764bi (A002260 n) (A004736 n)))
(define (A135764bi row col) (* (A000079 (- row 1)) (+ -1 col col)))
;; Antti Karttunen, Feb 03 2015

Formula

From Antti Karttunen, Feb 03 2015: (Start)
A(row, col) = 2^(row-1) * ((2*col)-1) = A000079(row-1) * A005408(col-1).
A(row,col) = A064989(A135765(row,A249746(col))).
A(row,col) = A(row+1,col)/2 [discarding the topmost row and halving the rest of terms gives the array back].
A(row,col) = A(row,col+1) - A000079(row) [discarding the leftmost column and subtracting 2^{row number} from the rest of terms gives the array back].
(End)
G.f.: ((2*x+1)*Sum_{i>=0} 2^i*x^(i*(i+1)/2) + 2*(1-2*x)*Sum_{i>=0} i*x^(i*(i+1)/2) + (1-6*x)*Sum_{i>=0} x^(i*(i+1)/2) - 1 - 2*x)*x/(1-2*x)^2. These sums are related to Jacobi theta functions. - Robert Israel, Feb 03 2015

Extensions

More terms from Sean A. Irvine, Nov 23 2010
Name amended and the illustration of array in the example section transposed by Antti Karttunen, Feb 03 2015

A249811 Permutation of natural numbers: a(n) = A249741(A001511(n), A003602(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 11, 24, 13, 20, 15, 10, 17, 26, 19, 34, 21, 32, 23, 48, 25, 38, 27, 54, 29, 44, 31, 12, 33, 50, 35, 64, 37, 56, 39, 76, 41, 62, 43, 84, 45, 68, 47, 120, 49, 74, 51, 94, 53, 80, 55, 90, 57, 86, 59, 114, 61, 92, 63, 16, 65, 98, 67, 124, 69, 104, 71, 118, 73, 110, 75, 144, 77, 116, 79, 142, 81
Offset: 1

Views

Author

Antti Karttunen, Nov 06 2014

Keywords

Comments

In the essence, a(n) tells which number in square array A249741 (the sieve of Eratosthenes minus 1) is at the same position where n is in array A135764, which is formed from odd numbers whose binary expansions are shifted successively leftwards on the successive rows. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e., a(2n+1) = 2n+1 for all n.
Equally: a(n) tells which number in array A114881 is at the same position where n is in the array A054582, as they are the transposes of above two arrays.

Links

Crossrefs

Inverse: A249812.
Cf. A000079, A000265, A001511, A003602, A005408, A006093, A007814, A054582, A083140, A083221, A249741.
Similar or related permutations: A249814 ("deep variant"), A246676, A249815, A114881, A209268, A249725, A249741.
Differs from A246676 for the first time at n=14, where a(14)=20, while
A246676(14)=26.

Programs

Formula

In the following formulas, A083221 and A249741 are interpreted as bivariate functions:
a(n) = A083221(A001511(n),A003602(n)) - 1 = A249741(A001511(n),A003602(n)).
As a composition of related permutations:
a(n) = A114881(A209268(n)).
a(n) = A249741(A249725(n)).
a(n) = A249815(A246676(n)).
Other identities. For all n >= 1 the following holds:
a(A000079(n-1)) = A006093(n).

A249813 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2 * a(A078898(n+1))) - 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 15, 32, 13, 10, 11, 64, 17, 128, 31, 14, 29, 256, 63, 12, 25, 18, 19, 512, 21, 1024, 127, 30, 33, 20, 255, 2048, 61, 26, 27, 4096, 57, 8192, 511, 22, 125, 16384, 23, 24, 49, 34, 35, 32768, 37, 28, 1023, 62, 41, 65536, 2047, 131072, 253, 58, 59, 36, 65, 262144, 39, 126, 509, 524288, 4095, 1048576, 121, 50, 51, 40, 53
Offset: 1

Views

Author

Antti Karttunen, Nov 06 2014

Keywords

Comments

This sequence is a "recursed variant" of A249812.
See also the comments at the inverse permutation A249814.

Links

Crossrefs

Inverse: A249814.
Similar or related permutations: A246683, A249812, A250243.
Cf. A000079, A006093, A055396, A078898.
Differs from A246683 for the first time at n=20, where a(20) = 14, while A246683(20) = 18.

Formula

a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2 * a(A078898(n+1))) - 1).
As a composition of other permutations:
a(n) = A246683(A250243(n)).
Other identities. For all n >= 1, the following holds:
a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back.]
a(A006093(n)) = A000079(n-1).

A249816 Permutation of natural numbers: a(n) = A246275(A055396(n+1), A078898(n+1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 26, 21, 22, 23, 24, 25, 20, 27, 28, 29, 30, 31, 44, 33, 34, 35, 36, 37, 32, 39, 40, 41, 42, 43, 80, 45, 46, 47, 48, 49, 74, 51, 52, 53, 124, 55, 62, 57, 58, 59, 60, 61, 38, 63, 54, 65, 66, 67, 134, 69, 70, 71, 72, 73, 50, 75, 76, 77, 78, 79, 98, 81, 82, 83
Offset: 1

Views

Author

Antti Karttunen, Nov 06 2014

Keywords

Comments

a(n) tells which number in square array A246275 is at the same position where n is in array A249741, the sieve of Eratosthenes minus 1. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e. a(2n+1) = 2n+1 for all n. Also, as the leftmost column in both arrays is primes minus one (A006093), they are also among the fixed points.
Equally: a(n) tells which number in array A246273 is at the same position where n is in the array A114881, as they are the transposes of above two arrays.

Links

Crossrefs

Inverse: A249815.
Cf. A005408, A006093, A055396, A078898, A246278.
Similar or related permutations: A250243 ("deep variant"), A246676, A249812, A249818, A246273, A246275, A114881, A249741.
Differs from A249815 and A250244 for the first time at n=32, where a(32) = 44, while A249815(32) = A250244(32) = 38.
Differs from A250244 for the first time at n=39, where a(39) = 39, while A250243(39) = 51.

Programs

Formula

a(n) = A246275(A055396(n+1), A078898(n+1)).
As a composition of other permutations:
a(n) = A246676(A249812(n)).
a(n) = A249818(n+1) - 1.
Other identities. For all n >= 1:
a(A005408(n-1)) = A005408(n-1) and a(A006093(n)) = A006093(n). [Fixes odd numbers and precedents of primes. Cf. comments above].

A249742 Inverse permutation to A249741.

Original entry on oeis.org

1, 3, 2, 6, 4, 10, 7, 5, 11, 15, 16, 21, 22, 8, 29, 28, 37, 36, 46, 12, 56, 45, 67, 9, 79, 17, 92, 55, 106, 66, 121, 23, 137, 13, 154, 78, 172, 30, 191, 91, 211, 105, 232, 38, 254, 120, 277, 14, 301, 47, 326, 136, 352, 18, 379, 57, 407, 153, 436, 171, 466, 68, 497, 24, 529, 190, 562, 80, 596, 210, 631, 231, 667, 93, 704, 19
Offset: 1

Views

Author

Antti Karttunen, Nov 15 2014

Keywords

Links

Crossrefs

Inverse: A249741.
Cf. A000124, A000217, A005408, A006093, A055396, A078898.
Similar or related permutations: A249725, A249812, A250252.
Differs from A246276 for the first time at n=20, where a(20) = 12, while A246276(20) = 17.

Programs

  • Scheme
    (define (A249742 n) (let ((x (A055396 (+ 1 n))) (y (A078898 (+ 1 n)))) (* (/ 1 2) (- (expt (+ x y) 2) x y y y -2))))

Formula

a(n) = 1 + ((((x+y)^2) - x - 3*y)/2), where x = A055396(n+1) and y = A078898(n+1).
As a composition of related permutations:
a(n) = A249725(A249812(n)).
Other identities.
For all n >= 0 the following holds:
a(A005408(n)) = A000124(n). [Maps odd numbers to central polygonal numbers.]
For all n >= 1 the following holds:
a(A006093(n)) = A000217(n). [Maps precedents of primes to triangular numbers.]

A250252 Inverse permutation to A114881.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 10, 5, 15, 11, 21, 16, 28, 9, 36, 22, 45, 29, 55, 14, 66, 37, 78, 8, 91, 20, 105, 46, 120, 56, 136, 27, 153, 13, 171, 67, 190, 35, 210, 79, 231, 92, 253, 44, 276, 106, 300, 12, 325, 54, 351, 121, 378, 19, 406, 65, 435, 137, 465, 154, 496, 77, 528, 26, 561, 172, 595, 90, 630, 191, 666, 211, 703, 104, 741, 18
Offset: 1

Views

Author

Antti Karttunen, Nov 15 2014

Keywords

Links

Crossrefs

Inverse: A114881.
Similar or related permutations: A209268, A249812.
Cf. A000124, A000217, A005408, A006093, A055396, A078898.
Differs from A246274 for the first time at n=20, where a(20) = 14, while
A246274(20) = 20.

Programs

  • Scheme
    (define (A250252 n) (let ((x (A078898 (+ 1 n))) (y (A055396 (+ 1 n)))) (* (/ 1 2) (- (expt (+ x y) 2) x y y y -2))))

Formula

a(n) = 1 + ((((x+y)^2) - x - 3*y)/2), where x = A078898(n+1) and y = A055396(n+1).
As a composition of related permutations:
a(n) = A209268(A249812(n)).
Other identities. For all n >= 1 the following holds:
a(A005408(n-1)) = A000217(n). [Maps odd numbers to triangular numbers.]
a(A006093(n)) = A000124(n-1). [Maps precedents of primes to central polygonal numbers.]
Showing 1-6 of 6 results.

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