|
%I #30 May 27 2020 09:55:43
%S 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,
%T 27,28,30,34,36,40,48,64,23,29,31,33,35,37,38,39,41,42,44,45,50,51,52,
%U 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
%N Irregular triangle where row n gives all terms k for which A064097(k) = n.
%C 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.
%C Any prime that appears on row n is 1 + {some term on row n-1}.
%C 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.
%C A001221(k) gives the number of terms on the row above that are immediate descendants of k.
%C A067513(k) gives the number of terms on the row below that lead to k.
%H Michael De Vlieger, <a href="/A334111/b334111.txt">Table of n, a(n) for n = 0..14422</a> (rows 0 <= n <= 17, flattened)
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e Rows 0-6 of the irregular table:
%e 0 | 1;
%e 1 | 2;
%e 2 | 3, 4;
%e 3 | 5, 6, 8;
%e 4 | 7, 9, 10, 12, 16;
%e 5 | 11, 13, 14, 15, 17, 18, 20, 24, 32;
%e 6 | 19, 21, 22, 25, 26, 27, 28, 30, 34, 36, 40, 48, 64;
%t f[n_] := Length@ NestWhileList[# - #/FactorInteger[#][[1, 1]] &, n, # != 1 &]; SortBy[ Range@70, f]
%t (* Second program *)
%t 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 *)
%o (PARI)
%o A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
%o A064097(n) = if(1==n,0,1+A064097(A060681(n)));
%o for(n=0, 10, for(k=1,2^n,if(A064097(k)==n, print1(k,", "))));
%Y Cf. A001221, A064097, A067513, A333123, A334144.
%Y Cf. A105017 (left edge), A000079 (right edge), A175125 (row lengths).
%Y Cf. also A058812, A334100.
%K nonn,look,tabf
%O 0,2
%A _Antti Karttunen_, _Michael De Vlieger_ and _Robert G. Wilson v_, Apr 14 2020
|
