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A133255 Triangle with a minimum occurrence of prime powers for which the least common multiple of the rows will give the terms in A003418.

%I A133255 #9 Mar 23 2025 08:39:05
%S A133255 1,1,1,1,1,2,1,1,2,3,1,1,4,3,1,1,1,4,3,1,5,1,1,4,3,1,5,1,1,1,4,3,1,5,
%T A133255 1,7,1,1,8,3,1,5,1,7,1,1,1,8,9,1,5,1,7,1,1,1,1,8,9,1,5,1,7,1,1,1,1,1,
%U A133255 8,9,1,5,1,7,1,1,1,11,1,1,8,9,1,5,1,7,1,1,1,11,1,1,1,8,9,1,5,1,7,1,1,1,11,1
%N A133255 Triangle with a minimum occurrence of prime powers for which the least common multiple of the rows will give the terms in A003418.
%C A133255 Checked up to 29th row. Similar to A133232 and A133233. In this table the prime powers with the same base are in the same column. A prime power occurs in the table: (base of prime power-1)*(the prime power).
%F A133255 T(n,k) = if k=1 then 1 elseif n-1>=(A089026(n-1))^0 and n-1<(A089026(n-1))^1 then (A089026(n-1))^0 elseif n-1>=(A089026(n-1))^1 and n-1<(A089026(n-1))^2 then (A089026(n-1))^1 elseif n-1>=(A089026(n-1))^2 and n-1<(A089026(n-1))^3 then (A089026(n-1))^2 elseif n-1>=(A089026(n-1))^3 and n-1<(A089026(n-1))^4 then (A089026(n-1))^3 elseif n-1>=(A089026(n-1))^4 and n-1<(A089026(n-1))^5 then (A089026(n-1))^4 else 1 (1<=k<=n) And so on, this formula needs to be expanded if one wants to make a bigger table. A089026(n-1) means that the index to that sequence is shifted in this formula so that the first term in A089026 is used in the second column of the table.
%e A133255 lcm{1}= 1
%e A133255 lcm{1,1} = 1
%e A133255 lcm{1,1,2} = 2
%e A133255 lcm{1,1,2,3} = 6
%e A133255 lcm{1,1,4,3,1} = 12
%e A133255 lcm{1,1,4,3,1,5} = 60
%e A133255 lcm{1,1,4,3,1,5,1} = 60
%e A133255 lcm{1,1,4,3,1,5,1,7} = 420
%e A133255 lcm{1,1,8,3,1,5,1,7,1} = 840
%e A133255 lcm{1,1,8,9,1,5,1,7,1,1} = 2520
%e A133255 1 = 1
%e A133255 1*1 = 1
%e A133255 1*1*2 = 2
%e A133255 1*1*2*3 = 6
%e A133255 1*1*4*3*1 = 12
%e A133255 1*1*4*3*1*5 = 60
%e A133255 1*1*4*3*1*5*1 = 60
%e A133255 1*1*4*3*1*5*1*7 = 420
%e A133255 1*1*8*3*1*5*1*7*1 = 840
%e A133255 1*1*8*9*1*5*1*7*1*1 = 2520
%o A133255 (Excel) =if(column()=1;1; if(and(row()-1>=(A089026(n-1))^0;row()-1<(A089026(n-1))^1);(A089026(n-1))^0; if(and(row()-1>=(A089026(n-1))^1;row()-1<(A089026(n-1))^2);(A089026(n-1))^1; if(and(row()-1>=(A089026(n-1))^2;row()-1<(A089026(n-1))^3);(A089026(n-1))^2; if(and(row()-1>=(A089026(n-1))^3;row()-1<(A089026(n-1))^4);(A089026(n-1))^3; if(and(row()-1>=(A089026(n-1))^4;row()-1<(A089026(n-1))^5);(A089026(n-1))^4; 1)))))) And so on.
%Y A133255 Cf. A089026, A003418, A000961.
%K A133255 nonn,tabl,uned
%O A133255 1,6
%A A133255 _Mats Granvik_, Oct 14 2007