A335430 Square array where row n lists all numbers k for which A331410(k) = n, read by falling antidiagonals.

1, 2, 3, 4, 6, 5, 8, 7, 9, 15, 16, 12, 10, 17, 25, 32, 14, 11, 19, 29, 73, 64, 24, 13, 27, 37, 75, 125, 128, 28, 18, 30, 45, 85, 145, 365, 256, 31, 20, 33, 50, 87, 149, 375, 625, 512, 48, 21, 34, 51, 89, 173, 425, 725, 1249, 1024, 56, 22, 35, 53, 95, 185, 435, 745, 1489, 3125, 2048, 62, 23, 38, 55, 101, 219, 445, 841, 1825, 3625, 6245

Offset

0,2

Comments

Array is read by descending antidiagonals with (n,k) = (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ... where A(n,k) is the (k+1)-th solution x to A331410(x) = n. The row indexing (n) starts from 0, and column indexing (k) also from 0.

For any odd prime p that appears on row n, p+1 appears on row n-1.

The e-th powers of the terms on row n form a subset of terms on row (e*n). More generally, a product of terms that occur on rows i_1, i_2, ..., i_k can be found at row (i_1 + i_2 + ... + i_k), because A331410 is completely additive.

Links

Table of n, a(n) for n=0..77.

Index entries for sequences that are permutations of the natural numbers

Example

The top left corner of the array:
  n\k |    0     1     2     3     4     5     6     7     8     9
------+----------------------------------------------------------------
   0  |    1,    2,    4,    8,   16,   32,   64,  128,  256,  512, ...
   1  |    3,    6,    7,   12,   14,   24,   28,   31,   48,   56, ...
   2  |    5,    9,   10,   11,   13,   18,   20,   21,   22,   23, ...
   3  |   15,   17,   19,   27,   30,   33,   34,   35,   38,   39, ...
   4  |   25,   29,   37,   45,   50,   51,   53,   55,   57,   58, ...
   5  |   73,   75,   85,   87,   89,   95,  101,  109,  111,  113, ...
   6  |  125,  145,  149,  173,  185,  219,  225,  250,  255,  261, ...
   7  |  365,  375,  425,  435,  445,  447,  449,  475,  493,  499, ...
   8  |  625,  725,  745,  841,  865,  925,  997, 1009, 1073, 1095, ...
   9  | 1249, 1489, 1825, 1875, 1993, 2017, 2117, 2125, 2175, 2225, ...
etc.

Prog

(PARI)
up_to = 78-1; \\ = binomial(12+1, 2)-1
memoA331410 = Map();
A331410(n) = if(1==n, 0, my(v=0); if(mapisdefined(memoA331410, n, &v), v, my(f=factor(n)); v = sum(k=1, #f~, if(2==f[k, 1], 0, f[k, 2]*(1+A331410(f[k, 1]+1)))); mapput(memoA331410, n, v); (v)));
memoA335430sq = Map();
A335430sq(n, k) = { my(v=0); if((0==k), v = -1, if(!mapisdefined(memoA335430sq, [n, k-1], &v), v = A335430sq(n, k-1))); for(i=1+v, oo, if(A331410(1+i)==n, mapput(memoA335430sq, [n, k], i); return(1+i))); };
A335430list(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] = A335430sq(col, (a-(col))))); (v); };
v335430 = A335430list(up_to);
A335430(n) = v335430[1+n];
for(n=0, up_to, print1(A335430(n), ", "));

Crossrefs

Cf. A331410.

Cf. A329662 (the leftmost column), A000079, A335431, A335882 (rows 0, 1 and 2).

Cf. also A334100 (an analogous array for the map k -> k - k/p), and A335910.

Sequence in context: A123503 A123717 A123718 * A121878 A167905 A285039

Adjacent sequences: A335427 A335428 A335429 * A335431 A335432 A335433

Keyword

nonn,tabl

Author

Antti Karttunen, Jun 28 2020

Status

approved