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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.
0
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, 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, 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
OFFSET
1,6
COMMENTS
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).
FORMULA
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 the that sequence is shifted in this formula so that the first term in A089026 is used in the second column of the table.
EXAMPLE
lcm{1}= 1
lcm{1,1} = 1
lcm{1,1,2} = 2
lcm{1,1,2,3} = 6
lcm{1,1,4,3,1} = 12
lcm{1,1,4,3,1,5} = 60
lcm{1,1,4,3,1,5,1} = 60
lcm{1,1,4,3,1,5,1,7} = 420
lcm{1,1,8,3,1,5,1,7,1} = 840
lcm{1,1,8,9,1,5,1,7,1,1} = 2520
1 = 1
1*1 = 1
1*1*2 = 2
1*1*2*3 = 6
1*1*4*3*1 = 12
1*1*4*3*1*5 = 60
1*1*4*3*1*5*1 = 60
1*1*4*3*1*5*1*7 = 420
1*1*8*3*1*5*1*7*1 = 840
1*1*8*9*1*5*1*7*1*1 = 2520
PROG
(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.
CROSSREFS
KEYWORD
nonn,tabl,uned
AUTHOR
Mats Granvik, Oct 14 2007
STATUS
approved