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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.

%I A328250 #17 Oct 13 2019 18:08:42
%S A328250 4,8,1,12,2,9,16,3,18,50,20,5,25,99,306,24,6,45,125,549,5831,27,7,49,
%T A328250 207,1611,6849,20230,28,10,63,343,2662,14225,33026,52283,32,11,75,375,
%U A328250 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
%N 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.
%C A328250 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.
%H A328250 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A328250 The upper left corner of the array:
%e A328250          4,        8,       12,       16,       20,       24,       27,       28,
%e A328250          1,        2,        3,        5,        6,        7,       10,       11,
%e A328250          9,       18,       25,       45,       49,       63,       75,       90,
%e A328250         50,       99,      125,      207,      343,      375,      531,      686,
%e A328250        306,      549,     1611,     2662,     2842,     2891,     4575,     4802,
%e A328250       5831,     6849,    14225,    16299,    19431,    21231,    22638,    24010,
%e A328250      20230,    33026,    47107,    49806,   117649,   121671,   145386,   162707,
%e A328250      52283,   225998,  1336047,  1422275,  1500759,  1576899,  2309503,  3023398,
%e A328250     286891,  1292750,  2886982,  3137526,  6882453,  8703459, 15358457, 16777114,
%e A328250   10820131, 21628098, 23934105, 24332763, 46295435, 51320698, 52320191, 56199375,
%e A328250   38452606, ...
%e A328250   ...
%o A328250 (PARI)
%o A328250 up_to = 45; \\ 10585 = binomial(145+1,2)
%o A328250 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));
%o A328250 A328248(n) = { my(k=1); while(n && !issquarefree(n), k++; n = A003415checked(n)); (!!n*k); };
%o A328250 memoA328250sq = Map();
%o A328250 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))); };
%o A328250 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); };
%o A328250 v328250 = A328250list(up_to);
%o A328250 A328250(n) = v328250[n];
%Y A328250 Cf. A003415, A328248.
%Y A328250 Column 1: A328302.
%Y A328250 Rows 1 - 4 are: A328251, A005117, A328252, A328253.
%K A328250 nonn,tabl
%O A328250 1,1
%A A328250 _Antti Karttunen_, Oct 12 2019