A328161 -id:A328161 - cp's OEIS frontend

A328171 Number of (necessarily strict) integer partitions of n with no two consecutive parts divisible.

1, 1, 1, 1, 1, 2, 1, 3, 2, 4, 4, 5, 4, 9, 9, 10, 12, 14, 16, 20, 23, 29, 34, 38, 41, 51, 60, 66, 78, 89, 103, 119, 137, 157, 180, 201, 229, 261, 298, 338, 379, 431, 486, 547, 618, 694, 783, 876, 986, 1103, 1241, 1387, 1551, 1728, 1932, 2148, 2395, 2664, 2963

Offset: 0

Gus Wiseman, Oct 11 2019

nonn

			The a(1) = 1 through a(15) = 10 partitions (A..F = 10..15):
  1  2  3  4  5   6  7   8   9    A    B   C    D    E     F
              32     43  53  54   64   65  75   76   86    87
                     52      72   73   74  543  85   95    96
                             432  532  83  732  94   A4    B4
                                       92       A3   B3    D2
                                                B2   653   654
                                                643  743   753
                                                652  752   852
                                                832  5432  A32
                                                           6432

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..330

The complement is counted by A328221.
The Heinz numbers of these partitions are A328603.
Partitions whose pairs of consecutive parts are relatively prime are A328172, with strict case A328188.
Partitions with no pair of consecutive parts relatively prime are A328187, with strict case A328220.
Numbers without consecutive divisible proper divisors are A328028.
Cf. A000837, A018783, A328026, A328161, A328189, A328194, A328195.

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

A328460 Number of compositions of n with no part divisible by the next.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 8, 11, 16, 26, 35, 53, 76, 115, 168, 244, 363, 528, 782, 1144, 1685, 2474, 3633, 5347, 7844, 11539, 16946, 24919, 36605, 53782, 79053, 116142, 170700, 250800, 368585, 541610, 795884, 1169572, 1718593, 2525522, 3711134, 5453542, 8013798, 11776138

Offset: 0

Gus Wiseman, Oct 17 2019

nonn

			The a(1) = 1 through a(9) = 16 compositions:
  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)     (9)
            (21)  (31)  (23)  (42)   (25)   (35)    (27)
                        (32)  (51)   (34)   (53)    (45)
                        (41)  (231)  (43)   (62)    (54)
                              (321)  (52)   (71)    (63)
                                     (61)   (251)   (72)
                                     (232)  (323)   (81)
                                     (421)  (341)   (234)
                                            (431)   (252)
                                            (521)   (342)
                                            (2321)  (351)
                                                    (423)
                                                    (432)
                                                    (531)
                                                    (621)
                                                    (3231)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The case of partitions is A328171.
If we also require no part to be divisible by the prior, we get A328508.
Compositions with each part relatively prime to the next are A167606.
Compositions with no part relatively prime to the next are A178470.
Cf. A328026, A328028, A328161, A328172, A328189.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,y_,_}/;Divisible[y,x]]&]],{n,0,10}]

seq(n)={my(r=matid(n)); for(k=1, n, for(i=1, k-1, r[i,k]=sum(j=1, k-i, if(j%i, r[j, k-i])))); concat([1], vecsum(Col(r)))} \\ Andrew Howroyd, Oct 19 2019

Terms a(26) and beyond from Andrew Howroyd, Oct 19 2019

A328028 Nonprime numbers n whose proper divisors (greater than 1 and less than n) have no consecutive divisibilities.

Original entry on oeis.org

1, 4, 6, 9, 10, 12, 14, 15, 21, 22, 24, 25, 26, 30, 33, 34, 35, 36, 38, 39, 45, 46, 48, 49, 51, 55, 57, 58, 60, 62, 63, 65, 69, 70, 72, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 96, 105, 106, 108, 111, 115, 118, 119, 120, 121, 122, 123, 129, 132, 133, 134

Offset: 1

Gus Wiseman, Oct 06 2019

nonn

			The proper divisors of 18 are {2, 3, 6, 9}, and {3, 6} are a consecutive divisible pair, so 18 does not belong to the sequence.
The proper divisors of 60 are {2, 3, 4, 5, 6, 10, 12, 15, 20, 30}, and none of {2, 3}, {3, 4}, {4, 5}, {5, 6}, {6, 10}, {10, 12}, {12, 15}, {15, 20}, or {20, 30} are divisible pairs, so 60 belongs to the sequence.

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

Positions of 0's or 2's in A328026.
1 and positions of 1's in A328194.
The version including primes is A328161.
Partitions with no consecutive divisibilities are A328171.
Numbers whose proper divisors have no consecutive successions are A088725.
Contains A001358.
Cf. A000005, A060680, A060775, A067513, A163870, A328162, A328189.

filter:= proc(n) local D,i;
  if isprime(n) then return false fi;
  D:= sort(convert(numtheory:-divisors(n) minus {1,n}, list));
  for i from 1 to nops(D)-1 do if (D[i+1]/D[i])::integer then return false fi od:
  true
end proc:
select(filter, [$1..300]); # Robert Israel, Oct 11 2019

Select[Range[100],!PrimeQ[#]&&!MatchQ[DeleteCases[Divisors[#],1|#],{_,x_,y_,_}/;Divisible[y,x]]&]

A328189 Numbers n with at least one pair of consecutive divisible nontrivial divisors (greater than 1 and less than n).

Original entry on oeis.org

8, 16, 18, 20, 27, 28, 32, 40, 42, 44, 50, 52, 54, 56, 64, 66, 68, 75, 76, 78, 80, 81, 88, 92, 98, 99, 100, 102, 104, 110, 112, 114, 116, 117, 124, 125, 126, 128, 130, 136, 138, 140, 147, 148, 152, 153, 156, 160, 162, 164, 170, 171, 172, 174, 176, 184, 186

Offset: 1

Gus Wiseman, Oct 13 2019

nonn

			The nontrivial divisors of 42 are {2, 3, 6, 7, 14, 21}, with pairs of consecutive divisible divisors {3, 6} and {7, 14}, so 42 belongs to the sequence.

Complement of A328161.
Positions of terms greater than 1 in A328194.
Partitions with a pair of consecutive divisible parts are A328221.
Cf. A000005, A060680, A060775, A067513, A088725, A163870, A328028, A328162, A328171.

Select[Range[200],MatchQ[DeleteCases[Divisors[#],1|#],{_,x_,y_,_}/;Divisible[y,x]]&]
Select[Range[2,200],AnyTrue[Partition[Most[Rest[Divisors[#]]],2,1],Mod[#[[2]],#[[1]]] == 0&]&] (* Harvey P. Dale, Mar 14 2023 *)

A328194 Maximum length of a divisibility chain of consecutive nontrivial divisors of n (greater than 1 and less than n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 1, 1, 3, 0, 2, 0, 2, 1, 1, 0, 1, 1, 1, 2, 2, 0, 1, 0, 4, 1, 1, 1, 1, 0, 1, 1, 2, 0, 2, 0, 2, 1, 1, 0, 1, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 1, 0, 1, 1, 5, 1, 2, 0, 2, 1, 1, 0, 1, 0, 1, 2, 2, 1, 2, 0, 2, 3, 1, 0, 1, 1, 1, 1, 3, 0, 1, 1, 2, 1, 1, 1, 1, 0, 2, 2, 3, 0, 2, 0, 3, 1

Offset: 1

Gus Wiseman, Oct 14 2019

nonn

The nontrivial divisors of n are row n of A163870.

			The nontrivial divisors of 272 are {2, 4, 8, 16, 17, 34, 68, 136} with divisibility chains {{2, 4, 8, 16}, {17, 34, 68, 136}}, so a(272) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Positions of 1's are A328028 without 1.
The version with all divisors allowed is A328162.
Allowing n as a divisor of n gives A328195.
Indices of terms greater than 1 are A328189.
The maximum run-length of divisors of n is A055874(n).
Cf. A000005, A033676, A060775, A163870, A328026, A328161, A328171.

Table[Switch[n,1,0,?PrimeQ,0,,Max@@Length/@Split[DeleteCases[Divisors[n],1|n],Divisible[#2,#1]&]],{n,100}]

A328194(n) = if(1==n || isprime(n), 0, my(divs=divisors(n), rl=0,ml=1); for(i=2,#divs-1,if(!(divs[i]%divs[i-1]), rl++, ml = max(rl,ml); rl=1)); max(ml,rl)); \\ Antti Karttunen, Dec 07 2024

Data section extended up to a(105) by Antti Karttunen, Dec 07 2024

A328508 Number of compositions of n with no part divisible by the next or the prior.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 6, 4, 8, 14, 14, 27, 30, 55, 69, 97, 155, 200, 312, 421, 630, 893, 1260, 1864, 2600, 3813, 5395, 7801, 11196, 15971, 23126, 32917, 47514, 67993, 97670, 140334, 200913, 289147, 414119, 595109, 853751, 1225086, 1759405, 2523151, 3623984, 5198759

Offset: 0

Gus Wiseman, Oct 17 2019

nonn

			The a(1) = 1 through a(11) = 14 compositions (A = 10, B = 11):
  (1)  (2)  (3)  (4)  (5)   (6)  (7)    (8)    (9)    (A)     (B)
                      (23)       (25)   (35)   (27)   (37)    (29)
                      (32)       (34)   (53)   (45)   (46)    (38)
                                 (43)   (323)  (54)   (64)    (47)
                                 (52)          (72)   (73)    (56)
                                 (232)         (234)  (235)   (65)
                                               (252)  (253)   (74)
                                               (432)  (325)   (83)
                                                      (343)   (92)
                                                      (352)   (254)
                                                      (523)   (272)
                                                      (532)   (353)
                                                      (2323)  (434)
                                                      (3232)  (452)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The case of partitions is A328171.
If we only forbid parts to be divisible by the next, we get A328460.
Compositions with each part relatively prime to the next are A167606.
Compositions with no part relatively prime to the next are A178470.
Cf. A328026, A328028, A328161, A328172, A328189.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,y_,_}/;Divisible[y,x]||Divisible[x,y]]&]],{n,0,10}]

seq(n)={my(r=matid(n)); for(k=1, n, for(i=1, k-1, r[i,k]=sum(j=1, k-i, if(i%j && j%i, r[j, k-i])))); concat([1], vecsum(Col(r)))} \\ Andrew Howroyd, Oct 19 2019

Terms a(26) and beyond from Andrew Howroyd, Oct 19 2019

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

Original entry on oeis.org

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}]

A328195 Maximum length of a divisibility chain of consecutive divisors of n greater than 1.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 4, 1, 2, 1, 3, 2, 2, 1, 2, 2, 2, 3, 3, 1, 2, 1, 5, 2, 2, 2, 2, 1, 2, 2, 3, 1, 2, 1, 3, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 2, 2, 1, 2, 1, 2, 2, 6, 2, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 3, 2, 2, 1, 3, 4, 2, 1, 2, 2, 2, 2, 4, 1, 2, 2, 3, 2, 2, 2, 2, 1, 2, 3, 3, 1, 2, 1, 4, 2

Offset: 1

Gus Wiseman, Oct 14 2019

nonn

Also the maximum length of a divisibility chain of consecutive divisors of n less than n.

The divisors of n (except 1) are row n of A027749.

			The divisors of 272 greater than 1 are {2, 4, 8, 16, 17, 34, 68, 136, 272}, with divisibility chains {{2, 4, 8, 16}, {17, 34, 68, 136, 272}}, so a(272) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Allowing 1 as a divisor gives A328162.
Forbidding n as a divisor of n gives A328194.
Positions of 1's are A000040 (primes).
Indices of terms greater than 1 are A002808 (composite numbers).
The maximum run-length of divisors of n is A055874(n).
Cf. A000005, A033676, A060775, A163870, A328026, A328161, A328171.

Table[If[n==1,0,Max@@Length/@Split[DeleteCases[Divisors[n],1],Divisible[#2,#1]&]],{n,100}]

A328195(n) = if(1==n, 0, my(divs=divisors(n), rl=0,ml=1); for(i=2,#divs,if(!(divs[i]%divs[i-1]), rl++, ml = max(rl,ml); rl=1)); max(ml,rl)); \\ Antti Karttunen, Dec 07 2024

Data section extended up to a(105) by Antti Karttunen, Dec 07 2024

cp's OEIS Frontend

A328171 Number of (necessarily strict) integer partitions of n with no two consecutive parts divisible.

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A328460 Number of compositions of n with no part divisible by the next.

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A328028 Nonprime numbers n whose proper divisors (greater than 1 and less than n) have no consecutive divisibilities.

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A328189 Numbers n with at least one pair of consecutive divisible nontrivial divisors (greater than 1 and less than n).

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A328194 Maximum length of a divisibility chain of consecutive nontrivial divisors of n (greater than 1 and less than n).

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A328508 Number of compositions of n with no part divisible by the next or the prior.

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

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Mathematica

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

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Maple