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 A168017 Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n listed in decreasing order. 7
 1, 1, 2, 1, 3, 1, 2, 5, 1, 7, 1, 2, 3, 11, 1, 15, 1, 2, 5, 22, 1, 3, 30, 1, 2, 7, 42, 1, 56, 1, 2, 3, 5, 11, 77, 1, 101, 1, 2, 15, 135, 1, 3, 7, 176, 1, 2, 5, 22, 231, 1, 297, 1, 2, 3, 11, 30, 385, 1, 490, 1, 2, 5, 7, 42, 627, 1, 3, 15, 792, 1, 2, 56, 1002 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Positive values of triangle A168016. The number of terms of row n is equal to the number of divisors of n: A000005(n). Note that the last term of each row is the number of partitions of n: A000041(n). Also, it appears that row n lists the partition numbers of the divisors of n. [Omar E. Pol, Nov 23 2009] LINKS Alois P. Heinz, Rows n = 1..1400, flattened Omar E. Pol, Illustration of the partitions of n, for n = 1 .. 9 EXAMPLE Consider row n=8: (1, 2, 5, 22). The divisors of 8 listed in decreasing order are 8, 4, 2, 1 (see A056538). There is 1 partition of 8 into parts divisible by 8. Also, there are 2 partitions of 8 into parts divisible by 4: {(8), (4+4)}; 5 partitions of 8 into parts divisible by 2: {(8), (6+2), (4+4), (4+2+2), (2+2+2+2)}; and 22 partitions of 8 into parts divisible by 1, because A000041(8)=22. Then row 8 is formed by 1, 2, 5, 22. Triangle begins: 1; 1,  2; 1,  3; 1,  2,  5; 1,  7; 1,  2,  3, 11; 1, 15; 1,  2,  5, 22; 1,  3, 30; 1,  2,  7, 42; 1, 56; 1,  2,  3,  5, 11, 77; MAPLE with(numtheory): b:= proc(n, i, d) option remember;       if n<0 then 0     elif n=0 then 1     elif i<1 then 0     else b(n, i-d, d) +b(n-i, i, d)       fi     end: T:= proc(n) local l;       l:= sort([divisors(n)[]], `>`);       seq(b(n, n, l[i]), i=1..nops(l))     end: seq(T(n), n=1..30); # Alois P. Heinz, Oct 21 2011 MATHEMATICA b[n_, i_, d_] := b[n, i, d] = Which[n<0, 0, n==0, 1, i<1, 0, True, b[n, i - d, d] + b[n-i, i, d]]; T[n_] := Module[{l = Divisors[n] // Reverse}, Table[b[n, n, l[[i]]], {i, 1, Length[l]}]]; Table[T[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, Dec 03 2015, after Alois P. Heinz *) CROSSREFS Row sums give A047968. Cf. A000005, A000041, A056538, A135010, A138121, A168016, A168018, A168019, A168020, A168021. Sequence in context: A080521 A169613 A176572 * A293980 A240694 A258643 Adjacent sequences:  A168014 A168015 A168016 * A168018 A168019 A168020 KEYWORD nonn,look,tabf AUTHOR Omar E. Pol, Nov 22 2009 STATUS approved

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Last modified October 15 23:54 EDT 2019. Contains 328038 sequences. (Running on oeis4.)