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A182869 Joint-rank array of prime powers: p(i)^j, i>=1, j>=1, read by antidiagonals.

%I A182869 #15 Oct 15 2024 23:16:57
%S A182869 1,3,2,6,7,4,10,15,14,5,18,32,42,23,8,27,68,136,86,41,9,44,152,482,
%T A182869 392,244,53,11,70,359,1880,2001,1773,360,91,12,117,893,7771,11211
%N A182869 Joint-rank array of prime powers:  p(i)^j, i>=1, j>=1, read by antidiagonals.
%C A182869 Joint-rank arrays are defined in the first comment at A182801.  A182869 is a permutation of the positive integers.
%F A182869 T(i,j) = Sum_{h>=1} floor(j*log(p(i))/log(p(h))), where p(i) denotes the i-th prime.
%e A182869 First, arrange the prime powers in rows:
%e A182869 2....4....8....16....32...
%e A182869 3....9...27....81...243...
%e A182869 5...25..125...625..3125...
%e A182869 Then replace each prime power by its rank when they are all jointly ranked:
%e A182869 1....3....6....10.....18...
%e A182869 2....7...15....32.....68...
%e A182869 4...14...42...136....482...
%e A182869 5...23...86...392...2001...
%e A182869 8...41..244..1773..14901...
%t A182869 T[i_,j_]:=Sum[Floor[j*Log[Prime[i]]/Log[Prime[h]]],{h,1,PrimePi[Prime[i]^j]}];
%t A182869 TableForm[Table[T[i,j],{i,1,6},{j,1,6}]]
%Y A182869 Cf. A000961, A027883, A182801, A182870, A182908.
%K A182869 nonn,tabl
%O A182869 1,2
%A A182869 _Clark Kimberling_, Dec 09 2010
%E A182869 Corrected and extended by _Clark Kimberling_, Dec 13 2010