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

Author

Antti Karttunen, Jun 28 2020

Keywords

• nonn
• tabl

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.

Examples

			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.
		

Links

• Index entries for sequences that are permutations of the natural numbers

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.

Programs

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),", "));