OFFSET
1,2
COMMENTS
Array is read by descending antidiagonals with (n,k) = (0,1), (0,2), (1,1), (0,3), (1,2), (2,1), ... where A(n,k) is the k-th solution x to A329697(x) = n. The row indexing (n) starts from 0, and column indexing (k) from 1.
Any odd prime that appears on row n is 1+{some term 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 A329697 is completely additive.
The binary weight (A000120) of any term on row n is at most 2^n.
LINKS
EXAMPLE
The top left corner of the array:
n\k | 1 2 3 4 5 6 7 8 9 10
------+----------------------------------------------------------------
0 | 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...
1 | 3, 5, 6, 10, 12, 17, 20, 24, 34, 40, ...
2 | 7, 9, 11, 13, 14, 15, 18, 22, 25, 26, ...
3 | 19, 21, 23, 27, 29, 31, 33, 35, 37, 38, ...
4 | 43, 47, 49, 57, 59, 63, 67, 69, 71, 77, ...
5 | 127, 129, 133, 139, 141, 147, 161, 163, 171, 173, ...
6 | 283, 301, 329, 343, 347, 361, 379, 381, 383, 387, ...
7 | 659, 817, 827, 839, 849, 863, 883, 889, 893, 903, ...
8 | 1319, 1699, 1787, 1849, 1977, 1979, 1981, 2021, 2039, 2083, ...
9 | 3957, 4079, 4613, 4903, 5097, 5179, 5361, 5377, 5399, 5419, ...
etc.
Note that the row 9 is the first one which begins with composite, as 3957 = 3*1319. The next such rows are row 15 and row 22. See A334099.
MATHEMATICA
Block[{nn = 16, s}, s = Values@ PositionIndex@ Array[-1 + Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] &, 2^nn]; Table[s[[#, k]] &[m - k + 1], {m, nn - Ceiling[nn/4]}, {k, m, 1, -1}]] // Flatten (* Michael De Vlieger, Apr 30 2020 *)
PROG
(PARI)
up_to = 105; \\ up_to = 1081; \\ = binomial(46+1, 2)
memoA334100sq = Map();
A334100sq(n, k) = { my(v=0); if(!mapisdefined(memoA334100sq, [n, k-1], &v), if(1==k, v=0, v = A334100sq(n, k-1))); for(i=1+v, oo, if(A329697(i)==(n-1), mapput(memoA334100sq, [n, k], i); return(i))); };
A334100list(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] = A334100sq(col, (a-(col-1))))); (v); };
v334100 = A334100list(up_to);
A334100(n) = v334100[n];
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, Apr 14 2020
STATUS
approved