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

Author

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

Keywords

• nonn
• look
• tabf

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.

Examples

			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;
		

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

Crossrefs

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

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

Cf. also A058812, A334100.

Programs

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 *)

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,", "))));