A334100 Square array where the row n lists all numbers k for which A329697(k) = n, read by falling antidiagonals.

1, 2, 3, 4, 5, 7, 8, 6, 9, 19, 16, 10, 11, 21, 43, 32, 12, 13, 23, 47, 127, 64, 17, 14, 27, 49, 129, 283, 128, 20, 15, 29, 57, 133, 301, 659, 256, 24, 18, 31, 59, 139, 329, 817, 1319, 512, 34, 22, 33, 63, 141, 343, 827, 1699, 3957, 1024, 40, 25, 35, 67, 147, 347, 839, 1787, 4079, 9227, 2048, 48, 26, 37, 69, 161, 361, 849, 1849, 4613, 9233, 21599

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

Table of n, a(n) for n=1..78.

Index entries for sequences that are permutations of the natural numbers

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)
A329697(n) = if(!bitand(n, n-1), 0, 1+A329697(n-(n/vecmax(factor(n)[, 1]))));
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];

Crossrefs

Cf. A329697.

Cf. A334099 (the leftmost column).

Cf. A000079, A334101, A334102, A334103, A334104, A334105, A334106 for the rows 0-6.

Cf. A019434, A334092, A334093, A334094, A334095, A334096 for the primes on the rows 1-6.

Cf. also irregular triangle A334111.

Sequence in context: A306010 A162375 A370793 * A194068 A194062 A085177

Adjacent sequences: A334097 A334098 A334099 * A334101 A334102 A334103

Keyword

nonn,tabl

Author

Antti Karttunen, Apr 14 2020

Status

approved