A168018 - OEIS

%I #12 Apr 02 2017 03:20:01

%S 1,2,1,3,1,5,2,1,7,1,11,3,2,1,15,1,22,5,2,1,30,3,1,42,7,2,1,56,1,77,

%T 11,5,3,2,1,101,1,135,15,2,1,176,7,3,1,231,22,5,2,1,297,1,385,30,11,3,

%U 2,1,490,1,627,42,7,5,2,1,792,15,3,1,1002,56,2,1,1255,1,1575,77,22,11,5,3,2

%N 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.

%C Positive values of triangle A168021.

%C Note that column 1 lists the numbers of partitions A000041(n).

%C Row n has A000005(n) terms.

%C Also, it appears that row n lists the partition numbers of the divisors of n, in decreasing order. [_Omar E. Pol_, Nov 23 2009]

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the partitions of n, for n = 1 .. 9</a>

%e For example:

%e Consider row 8: (22, 5, 2, 1). The divisors of 8 are 1, 2, 4, 8 (see A027750). Also, there are 22 partitions of 8 into parts divisible by 1 (A000041(8)=22); 5 partitions of 8 into parts divisible by 2: {(8),(6+2),(4+4),(4+2+2),(2+2+2+2)}; 2 partitions of 8 into parts divisible by 4: {(8),(4+4)}; and 1 partition of 8 into parts divisible by 8. Then row 8 is formed by 22, 5, 2, 1.

%e Triangle begins:

%e 1;

%e 2, 1;

%e 3, 1;

%e 5, 2, 1;

%e 7, 1;

%e 11, 3, 2, 1;

%e 15, 1;

%e 22, 5, 2, 1;

%e 30, 3, 1;

%e 42, 7, 2, 1;

%e 56, 1;

%e 77, 11, 5, 3, 2, 1;

%p A168018 := proc(n) local dvs,p,i,d,a,pp,divs,par; dvs := sort(convert(numtheory[divisors](n),list)) ; p := combinat[partition](n) ; for i from 1 to nops(dvs) do d := op(i,dvs) ; a := 0 ; for pp in p do divs := true; for par in pp do if par mod d <> 0 then divs := false; end if; end do ; if divs then a := a+1 ; end if; end do ; printf("%d,",a) ; end do ; end proc: for n from 1 to 40 do A168018(n) ; end do : # _R. J. Mathar_, Feb 05 2010

%Y Row sums give A047968.

%Y Cf. A000005, A000041, A027750, A135010, A138121, A168016, A168017, A168019, A168020, A168021.

%K nonn,tabf

%O 1,2

%A _Omar E. Pol_, Nov 22 2009

%E Terms beyond row 12 from _R. J. Mathar_, Feb 05 2010