A328250 Square array A(n,k) read by descending antidiagonals where A(n,k) is the k-th solution x to A328248(x) = n-1.

4, 8, 1, 12, 2, 9, 16, 3, 18, 50, 20, 5, 25, 99, 306, 24, 6, 45, 125, 549, 5831, 27, 7, 49, 207, 1611, 6849, 20230, 28, 10, 63, 343, 2662, 14225, 33026, 52283, 32, 11, 75, 375, 2842, 16299, 47107, 225998, 286891, 36, 13, 90, 531, 2891, 19431, 49806, 1336047, 1292750, 10820131, 40, 14, 98, 686, 4575, 21231, 117649, 1422275, 2886982, 21628098, 38452606

Offset: 1

Author

Antti Karttunen, Oct 12 2019

Keywords

• nonn
• tabl

Comments

Row 1 of the array is reserved for numbers for which no squarefree number is ever reached, and from then on, each row n > 1 of array gives in ascending order all natural numbers that require n-2 iterations of arithmetic derivative (A003415) to reach a squarefree number. Squarefree numbers (A005117) thus occupy the row 2, as they require no iterations.

Examples

			The upper left corner of the array:
         4,        8,       12,       16,       20,       24,       27,       28,
         1,        2,        3,        5,        6,        7,       10,       11,
         9,       18,       25,       45,       49,       63,       75,       90,
        50,       99,      125,      207,      343,      375,      531,      686,
       306,      549,     1611,     2662,     2842,     2891,     4575,     4802,
      5831,     6849,    14225,    16299,    19431,    21231,    22638,    24010,
     20230,    33026,    47107,    49806,   117649,   121671,   145386,   162707,
     52283,   225998,  1336047,  1422275,  1500759,  1576899,  2309503,  3023398,
    286891,  1292750,  2886982,  3137526,  6882453,  8703459, 15358457, 16777114,
  10820131, 21628098, 23934105, 24332763, 46295435, 51320698, 52320191, 56199375,
  38452606, ...
  ...
		

Links

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

Crossrefs

Cf. A003415, A328248.

Column 1: A328302.

Rows 1 - 4 are: A328251, A005117, A328252, A328253.

Programs

PARI

up_to = 45; \\ 10585 = binomial(145+1,2)
A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
A328248(n) = { my(k=1); while(n && !issquarefree(n), k++; n = A003415checked(n)); (!!n*k); };
memoA328250sq = Map();
A328250sq(n, k) = { my(v=0); if(!mapisdefined(memoA328250sq,[n,k-1],&v),if(1==k, v=0, v = A328250sq(n, k-1))); for(i=1+v,oo,if((1+A328248(i))==n,mapput(memoA328250sq,[n,k],i); return(i))); };
A328250list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A328250sq(col,(a-(col-1))))); (v); };
v328250 = A328250list(up_to);
A328250(n) = v328250[n];