A328028 -id:A328028 - cp's OEIS frontend

A328221 Number of integer partitions of n with at least one pair of consecutive divisible parts.

0, 0, 1, 2, 4, 5, 10, 12, 20, 26, 38, 51, 73, 92, 126, 166, 219, 283, 369, 470, 604, 763, 968, 1217, 1534, 1907, 2376, 2944, 3640, 4476, 5501, 6723, 8212, 9986, 12130, 14682, 17748, 21376, 25717, 30847, 36959, 44152, 52688, 62714, 74557, 88440, 104775, 123878

Offset: 0

Gus Wiseman, Oct 15 2019

nonn

Includes all non-strict partitions.

			The a(2) = 1 through a(8) = 20 partitions:
  (11)  (21)   (22)    (41)     (33)      (61)       (44)
        (111)  (31)    (221)    (42)      (322)      (62)
               (211)   (311)    (51)      (331)      (71)
               (1111)  (2111)   (222)     (421)      (332)
                       (11111)  (321)     (511)      (422)
                                (411)     (2221)     (431)
                                (2211)    (3211)     (521)
                                (3111)    (4111)     (611)
                                (21111)   (22111)    (2222)
                                (111111)  (31111)    (3221)
                                          (211111)   (3311)
                                          (1111111)  (4211)
                                                     (5111)
                                                     (22211)
                                                     (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

The complement is counted by A328171.
Partitions whose consecutive parts are relatively prime are A328172.
Partitions with no pair of consecutive parts relatively prime are A328187.
Numbers without consecutive divisible proper divisors are A328028.
Cf. A000837, A018783, A328026, A328161, A328188, A328189, A328194, A328220.

Table[Length[Select[IntegerPartitions[n],MatchQ[#,{_,x_,y_,_}/;Divisible[x,y]]&]],{n,0,30}]

A328162 Maximum length of a divisibility chain of consecutive divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 4, 3, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 7, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 5, 2, 2, 2, 2, 2, 2

Offset: 1

Gus Wiseman, Oct 06 2019

nonn

			The divisors of 968 split into consecutive divisibility chains {{1, 2, 4, 8}, {11, 22, 44, 88}, {121, 242, 484, 968}}, so a(968) = 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Records occur at powers of 2 (A000079).
Taking only proper divisors gives A328194.
Taking only divisors > 1 gives A328195.
The maximum run-length among divisors of n is A055874.
Cf. A000005, A027750, A060680, A060775, A328026, A328028, A328161, A328189.

f:= proc(n) local F,L,d,i;
  F:= sort(convert(numtheory:-divisors(n),list));
  d:= nops(F);
  L:= Vector(d);
  L[1]:= 1;
  for i from 2 to d do
    if F[i] mod F[i-1] = 0 then L[i]:= L[i-1]+1
    else L[i]:= 1
    fi
  od;
  max(L)
end proc:
map(f, [$1..100]); # Robert Israel, Apr 20 2023

Table[Max@@Length/@Split[Divisors[n],Divisible[#2,#1]&],{n,100}]

cp's OEIS Frontend

A328221 Number of integer partitions of n with at least one pair of consecutive divisible parts.

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