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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.
6
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
1,1
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.
EXAMPLE
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, ...
...
PROG
(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];
CROSSREFS
Column 1: A328302.
Rows 1 - 4 are: A328251, A005117, A328252, A328253.
Sequence in context: A196618 A294830 A248415 * A340423 A367416 A295086
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, Oct 12 2019
STATUS
approved