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A335430
Square array where row n lists all numbers k for which A331410(k) = n, read by falling antidiagonals.
7
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
0,2
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.
EXAMPLE
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.
PROG
(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), ", "));
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.
Sequence in context: A123503 A123717 A123718 * A121878 A167905 A285039
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, Jun 28 2020
STATUS
approved