cp's OEIS Frontend

A341520 Square array A(n,k) = A156552(A005940(1+n)*A005940(1+k)), read by antidiagonals.

%I A341520 #38 Feb 28 2022 07:56:25
%S A341520 0,1,1,2,3,2,3,5,5,3,4,7,6,7,4,5,9,11,11,9,5,6,11,10,15,10,11,6,7,13,
%T A341520 13,19,19,13,13,7,8,15,14,23,12,23,14,15,8,9,17,23,27,21,21,27,23,17,
%U A341520 9,10,19,18,31,22,27,22,31,18,19,10,11,21,21,35,39,29,29,39,35,21,21,11,12,23,22,39,20,47,30,47,20,39,22,23,12
%N A341520 Square array A(n,k) = A156552(A005940(1+n)*A005940(1+k)), read by antidiagonals.
%C A341520 The indices run as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), etc. The array is symmetric.
%C A341520 This array defines a binary operation on the nonnegative integers that matches up the zeros in the binary representation of each operand (starting from the right, and including as many leading zeros as necessary) and concatenates the two (possibly null) strings of ones to the right of each matched pair of zeros. See the examples. - _Peter Munn_, Feb 14 2021.
%C A341520 As such it could be useful for implementing multiplication, say, in Turing machines, with a "tape-like" unary-binary encoding of the prime factorization of n (A156552). However, such representation is not very useful if addition or subtraction is also needed.
%H A341520 Antti Karttunen, <a href="/A341520/b341520.txt">Table of n, a(n) for n = 0..10439; the first 144 antidiagonals of the array</a>
%H A341520 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A341520 A(x, y) = A156552(A005940(1+x) * A005940(1+y)).
%F A341520 For all n>=0, A(0, n) = A(n, 0) = n.
%F A341520 For all x>=0, y>=0, A(x, y) = A(y, x).
%F A341520 For all x, y, z >= 0, A(x, A(y, z)) = A(A(x, y), z).
%F A341520 From _Antti Karttunen_, Feb 27 2022: (Start)
%F A341520 For all x, y >= 0, A(x, y) = A(A351961(x,y), A351962(x,y)).
%F A341520 For x >= 0, y > 0, A(x, y) = A351960(x, A(x, A297164(y))).
%F A341520 (End)
%e A341520 The top left {0..15} X {0..16} corner of the array:
%e A341520    0,  1,  2,  3,  4,  5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15,
%e A341520    1,  3,  5,  7,  9, 11,  13,  15,  17,  19,  21,  23,  25,  27,  29,  31,
%e A341520    2,  5,  6, 11, 10, 13,  14,  23,  18,  21,  22,  27,  26,  29,  30,  47,
%e A341520    3,  7, 11, 15, 19, 23,  27,  31,  35,  39,  43,  47,  51,  55,  59,  63,
%e A341520    4,  9, 10, 19, 12, 21,  22,  39,  20,  25,  26,  43,  28,  45,  46,  79,
%e A341520    5, 11, 13, 23, 21, 27,  29,  47,  37,  43,  45,  55,  53,  59,  61,  95,
%e A341520    6, 13, 14, 27, 22, 29,  30,  55,  38,  45,  46,  59,  54,  61,  62, 111,
%e A341520    7, 15, 23, 31, 39, 47,  55,  63,  71,  79,  87,  95, 103, 111, 119, 127,
%e A341520    8, 17, 18, 35, 20, 37,  38,  71,  24,  41,  42,  75,  44,  77,  78, 143,
%e A341520    9, 19, 21, 39, 25, 43,  45,  79,  41,  51,  53,  87,  57,  91,  93, 159,
%e A341520   10, 21, 22, 43, 26, 45,  46,  87,  42,  53,  54,  91,  58,  93,  94, 175,
%e A341520   11, 23, 27, 47, 43, 55,  59,  95,  75,  87,  91, 111, 107, 119, 123, 191,
%e A341520   12, 25, 26, 51, 28, 53,  54, 103,  44,  57,  58, 107,  60, 109, 110, 207,
%e A341520   13, 27, 29, 55, 45, 59,  61, 111,  77,  91,  93, 119, 109, 123, 125, 223,
%e A341520   14, 29, 30, 59, 46, 61,  62, 119,  78,  93,  94, 123, 110, 125, 126, 239,
%e A341520   15, 31, 47, 63, 79, 95, 111, 127, 143, 159, 175, 191, 207, 223, 239, 255,
%e A341520   16, 33, 34, 67, 36, 69,  70, 135,  40,  73,  74, 139,  76, 141, 142, 271,
%e A341520 ...
%e A341520 From _Peter Munn_, Feb 24 2021: (Start)
%e A341520 We consider the case of n = 10, k = 41, following the procedure in the Feb 14 2021 comment.
%e A341520 First, write 10 and 41 in binary:
%e A341520   10 = 1010_2
%e A341520   41 = 101001_2
%e A341520 Add at least one leading zero to each number, equalizing number of zeros:
%e A341520   0  0  1  0  1  0
%e A341520   0  1  0  1  0  0  1
%e A341520 Align zeros, but separate ones:
%e A341520   0     0  1     0  1  0
%e A341520   |     |        |     |
%e A341520   0  1  0     1  0     0  1
%e A341520 ---------------------------
%e A341520   0  1  0  1  1  0  1  0  1
%e A341520 Concatenating the ones, as shown above, we get 10110101_2 = 181.
%e A341520 So A(10, 41) = 181.
%e A341520 (End)
%t A341520 Block[{nn = 12, a = {1}}, Do[AppendTo[a, If[EvenQ[i], Times @@ Map[Prime[PrimePi[#1] + 1]^#2 & @@ # &, FactorInteger[#]] &@ a[[(i/2) + 1]], 2 a[[((i - 1)/2) + 1]]]], {i, nn}]; Table[Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &[a[[1 + n - k]]*a[[1 + k]] ], {n, 0, nn}, {k, n, 0, -1}]] // Flatten (* _Michael De Vlieger_, Feb 24 2021 *)
%o A341520 (PARI)
%o A341520 up_to = 105;
%o A341520 A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
%o A341520 A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
%o A341520 A341520sq(n,k) = A156552(A005940(1+n)*A005940(1+k));
%o A341520 A341520list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0,oo, for(col=0,a, i++; if(i > #v, return(v)); v[i] = A341520sq(col,(a-(col))))); (v); };
%o A341520 v341520 = A341520list(up_to);
%o A341520 A341520(n) = v341520[1+n];
%Y A341520 Cf. A005940, A156552.
%Y A341520 Cf. A088698 (main diagonal).
%Y A341520 Rows/columns 0-3: A001477, A005408, A341522, A004767. Row/column 7: A004771.
%Y A341520 Cf. A341521 (the lower triangular section).
%Y A341520 Cf. also A003991, A268725, A297164, A331590, A341509, A341510, A351960, A351961, A351962
%K A341520 nonn,tabl
%O A341520 0,4
%A A341520 _Antti Karttunen_, Feb 13 2021