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A334111
Irregular triangle where row n gives all terms k for which A064097(k) = n.
9
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.
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. A105017 (left edge), A000079 (right edge), A175125 (row lengths).
Cf. also A058812, A334100.
Sequence in context: A242704 A362178 A358122 * A243571 A215366 A333483
KEYWORD
nonn,look,tabf
STATUS
approved