

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
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OFFSET

0,2


COMMENTS

Applying map k > (p1)*(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 n1.
Any prime that appears on row n is 1 + {some term on row n1}.
The eth 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 06 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) = (nif(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: A332073 A117332 A242704 * 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



