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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.

%I A135764 #123 Jan 29 2025 08:03:32
%S A135764 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,
%T A135764 96,64,15,26,44,72,112,160,192,128,17,30,52,88,144,224,320,384,256,19,
%U A135764 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
%N 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.
%C A135764 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].
%C A135764 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
%C A135764 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
%C A135764 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
%C A135764 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
%C A135764 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
%C A135764 From _Gary W. Adamson_, Dec 30 2021: (Start)
%C A135764 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:
%C A135764   1   3   5  15
%C A135764   2   6  10  30
%C A135764   4  12  20  60. (End)
%H A135764 Antti Karttunen, <a href="/A135764/b135764.txt">Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array</a>
%H A135764 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A135764 From _Antti Karttunen_, Feb 03 2015: (Start)
%F A135764 A(row, col) = 2^(row-1) * ((2*col)-1) = A000079(row-1) * A005408(col-1).
%F A135764 A(row,col) = A064989(A135765(row,A249746(col))).
%F A135764 A(row,col) = A(row+1,col)/2 [discarding the topmost row and halving the rest of terms gives the array back].
%F A135764 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].
%F A135764 (End)
%F A135764 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
%e A135764 The table begins
%e A135764    1,  3,   5,   7,   9,  11,  13,  15,  17,  19,  21,  23, ...
%e A135764    2,  6,  10,  14,  18,  22,  26,  30,  34,  38,  42,  46, ...
%e A135764    4, 12,  20,  28,  36,  44,  52,  60,  68,  76,  84,  92, ...
%e A135764    8, 24,  40,  56,  72,  88, 104, 120, 136, 152, 168, 184, ...
%e A135764   16, 48,  80, 112, 144, 176, 208, 240, 272, 304, 336, 368, ...
%e A135764   32, 96, 160, 224, 288, 352, 416, 480, 544, 608, 672, 736, ...
%e A135764 etc.
%e A135764 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.
%e A135764 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.
%e A135764 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.
%p A135764 seq(seq(2^(j-1)*(2*(i-j)+1),j=1..i),i=1..20); # _Robert Israel_, Feb 03 2015
%t A135764 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 *)
%o A135764 (Scheme)
%o A135764 (define (A135764 n) (A135764bi (A002260 n) (A004736 n)))
%o A135764 (define (A135764bi row col) (* (A000079 (- row 1)) (+ -1 col col)))
%o A135764 ;; _Antti Karttunen_, Feb 03 2015
%o A135764 (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
%Y A135764 Transpose: A054582.
%Y A135764 Inverse permutation: A249725.
%Y A135764 Column 1: A000079.
%Y A135764 Row 1: A005408.
%Y A135764 Cf. A001511 (row index), A003602 (column index, both one-based).
%Y A135764 Related arrays: A135765, A253551, A254053, A254055.
%Y A135764 Cf. A000027, A002260, A004736, A064989, A135766, A249746.
%Y A135764 Cf. also permutations A246675, A246676, A249741, A249811, A249812.
%Y A135764 Cf. A260390.
%Y A135764 Cf. A001146, A051179.
%K A135764 easy,nonn,tabl
%O A135764 1,2
%A A135764 _Alford Arnold_, Nov 29 2007
%E A135764 More terms from _Sean A. Irvine_, Nov 23 2010
%E A135764 Name amended and the illustration of array in the example section transposed by _Antti Karttunen_, Feb 03 2015