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

%I A335430 #14 Jul 11 2020 02:49:24
%S A335430 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,
%T A335430 75,125,128,28,18,30,45,85,145,365,256,31,20,33,50,87,149,375,625,512,
%U A335430 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
%N A335430 Square array where row n lists all numbers k for which A331410(k) = n, read by falling antidiagonals.
%C A335430 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.
%C A335430 For any odd prime p that appears on row n, p+1 appears on row n-1.
%C A335430 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.
%H A335430 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A335430 The top left corner of the array:
%e A335430   n\k |    0     1     2     3     4     5     6     7     8     9
%e A335430 ------+----------------------------------------------------------------
%e A335430    0  |    1,    2,    4,    8,   16,   32,   64,  128,  256,  512, ...
%e A335430    1  |    3,    6,    7,   12,   14,   24,   28,   31,   48,   56, ...
%e A335430    2  |    5,    9,   10,   11,   13,   18,   20,   21,   22,   23, ...
%e A335430    3  |   15,   17,   19,   27,   30,   33,   34,   35,   38,   39, ...
%e A335430    4  |   25,   29,   37,   45,   50,   51,   53,   55,   57,   58, ...
%e A335430    5  |   73,   75,   85,   87,   89,   95,  101,  109,  111,  113, ...
%e A335430    6  |  125,  145,  149,  173,  185,  219,  225,  250,  255,  261, ...
%e A335430    7  |  365,  375,  425,  435,  445,  447,  449,  475,  493,  499, ...
%e A335430    8  |  625,  725,  745,  841,  865,  925,  997, 1009, 1073, 1095, ...
%e A335430    9  | 1249, 1489, 1825, 1875, 1993, 2017, 2117, 2125, 2175, 2225, ...
%e A335430 etc.
%o A335430 (PARI)
%o A335430 up_to = 78-1; \\ = binomial(12+1,2)-1
%o A335430 memoA331410 = Map();
%o A335430 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)));
%o A335430 memoA335430sq = Map();
%o A335430 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))); };
%o A335430 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); };
%o A335430 v335430 = A335430list(up_to);
%o A335430 A335430(n) = v335430[1+n];
%o A335430 for(n=0,up_to,print1(A335430(n),", "));
%Y A335430 Cf. A331410.
%Y A335430 Cf. A329662 (the leftmost column), A000079, A335431, A335882 (rows 0, 1 and 2).
%Y A335430 Cf. also A334100 (an analogous array for the map k -> k - k/p), and A335910.
%K A335430 nonn,tabl
%O A335430 0,2
%A A335430 _Antti Karttunen_, Jun 28 2020