A334111 Irregular triangle where row n gives all terms k for which A064097(k) = n.

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 13, 14, 15, 17, 18, 20, 24, 32, 19, 21, 22, 25, 26, 27, 28, 30, 34, 36, 40, 48, 64, 23, 29, 31, 33, 35, 37, 38, 39, 41, 42, 44, 45, 50, 51, 52, 54, 56, 60, 68, 72, 80, 96, 128, 43, 46, 49, 53, 55, 57, 58, 61, 62, 63, 65, 66, 70, 73, 74, 75, 76, 78, 81, 82, 84

Offset

0,2

Comments

Applying map k -> (p-1)*(k/p) to any term k on any row n > 1, where p is any prime factor of k, gives one of the terms on preceding row n-1.

Any 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 A064097 is completely additive.

A001221(k) gives the number of terms on the row above that are immediate descendants of k.

A067513(k) gives the number of terms on the row below that lead to k.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..14422 (rows 0 <= n <= 17, flattened)

Index entries for sequences that are permutations of the natural numbers

Example

Rows 0-6 of the irregular table:
0 |   1;
1 |   2;
2 |   3, 4;
3 |   5, 6, 8;
4 |   7, 9, 10, 12, 16;
5 |  11, 13, 14, 15, 17, 18, 20, 24, 32;
6 |  19, 21, 22, 25, 26, 27, 28, 30, 34, 36, 40, 48, 64;

Mathematica

f[n_] := Length@ NestWhileList[# - #/FactorInteger[#][[1, 1]] &, n, # != 1 &]; SortBy[ Range@70, f]
(* Second program *)
With[{nn = 8}, Values@ Take[KeySort@ PositionIndex@ Array[-1 + Length@ NestWhileList[# - #/FactorInteger[#][[1, 1]] &, #, # > 1 &] &, 2^nn], nn + 1]] // Flatten (* Michael De Vlieger, Apr 18 2020 *)

Prog

(PARI)
A060681(n) = (n-if(1==n, n, n/vecmin(factor(n)[, 1])));
A064097(n) = if(1==n, 0, 1+A064097(A060681(n)));
for(n=0, 10, for(k=1, 2^n, if(A064097(k)==n, print1(k, ", "))));

Crossrefs

Cf. A001221, A064097, A067513, A333123, A334144.

Cf. A105017 (left edge), A000079 (right edge), A175125 (row lengths).

Cf. also A058812, A334100.

Sequence in context: A242704 A362178 A358122 * A243571 A215366 A333483

Adjacent sequences: A334108 A334109 A334110 * A334112 A334113 A334114

Keyword

nonn,look,tabf

Author

Antti Karttunen, Michael De Vlieger and Robert G. Wilson v, Apr 14 2020

Status

approved