A328028 -id:A328028 - 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}]

A328172 Number of integer partitions of n with all pairs of consecutive parts relatively prime.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 10, 12, 16, 19, 24, 28, 36, 43, 51, 62, 74, 87, 104, 122, 143, 169, 195, 227, 260, 302, 346, 397, 455, 521, 599, 686, 780, 889, 1001, 1138, 1286, 1454, 1638, 1846, 2076, 2330, 2614, 2929, 3280, 3666, 4093, 4565, 5085, 5667, 6300, 7002

Offset: 0

Gus Wiseman, Oct 12 2019

nonn

Except for any number of 1's, these partitions must be strict. The fully strict case is A328188.

Partitions with no consecutive pair of parts relatively prime are A328187, with strict case A328220.

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (31)    (32)     (51)      (43)       (53)
             (111)  (211)   (41)     (321)     (52)       (71)
                    (1111)  (311)    (411)     (61)       (431)
                            (2111)   (3111)    (511)      (521)
                            (11111)  (21111)   (3211)     (611)
                                     (111111)  (4111)     (5111)
                                               (31111)    (32111)
                                               (211111)   (41111)
                                               (1111111)  (311111)
                                                          (2111111)
                                                          (11111111)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The case of compositions is A167606.
The strict case is A328188.
The Heinz numbers of these partitions are given by A328335.
Cf. A000837, A018783, A178470, A328028, A328170, A328171, A328187, A328188 A328220.

b:= proc(n, i, s) option remember; `if`(n=0 or i=1, 1,
      `if`(andmap(j-> igcd(i, j)=1, s), b(n-i, min(n-i, i-1),
           numtheory[factorset](i)), 0)+b(n, i-1, s))
    end:
a:= n-> b(n$2, {}):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 13 2019

Table[Length[Select[IntegerPartitions[n],!MatchQ[#,{_,x_,y_,_}/;GCD[x,y]>1]&]],{n,0,30}]
(* Second program: *)
b[n_, i_, s_] := b[n, i, s] = If[n == 0 || i == 1, 1,
     If[AllTrue[s, GCD[i, #] == 1&], b[n - i, Min[n - i, i - 1],
     FactorInteger[i][[All, 1]]], 0] + b[n, i - 1, s]];
a[n_] := b[n, n, {}];
a /@ Range[0, 60] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

A328187 Number of integer partitions of n with no pair of consecutive parts relatively prime.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 1, 5, 3, 8, 1, 14, 1, 16, 9, 22, 3, 38, 4, 46, 19, 58, 9, 94, 18, 106, 41, 144, 28, 221, 37, 246, 92, 318, 87, 465, 95, 530, 198, 693, 169, 963, 220, 1108, 424, 1383, 381, 1899, 492, 2216, 815, 2732, 799, 3644, 1041, 4231, 1585, 5194, 1608

Offset: 0

Gus Wiseman, Oct 12 2019

nonn

			The a(1) = 1 through a(15) = 9 partitions (A..F = 10..15):
  1  2  3  4   5  6    7  8     9    A      B  C       D  E        F
           22     33      44    63   55        66         77       96
                  42      62    333  64        84         86       A5
                  222     422        82        93         A4       C3
                          2222       442       A2         C2       555
                                     622       444        644      663
                                     4222      633        662      933
                                     22222     642        842      6333
                                               822        A22      33333
                                               3333       4442
                                               4422       6422
                                               6222       8222
                                               42222      44222
                                               222222     62222
                                                          422222
                                                          2222222

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..300

The Heinz numbers of these partitions are given by A328336.
The case of compositions is A178470.
The strict case is A328220.
Partitions with all pairs of consecutive parts relatively prime are A328172.
Cf. A000837, A018783, A328028, A328170, A328171, A328188.

Table[Length[Select[IntegerPartitions[n],!MatchQ[#,{_,x_,y_,_}/;GCD[x,y]==1]&]],{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

A328026 Number of divisible pairs of consecutive divisors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 03 2019

nonn

The number m = 2^n, n >= 0, is the smallest for which a(m) = n. - Marius A. Burtea, Nov 20 2019

			The divisors of 500 are {1,2,4,5,10,20,25,50,100,125,250,500}, with consecutive divisible pairs {1,2}, {2,4}, {5,10}, {10,20}, {25,50}, {50,100}, {125,250}, {250,500}, so a(500) = 8.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Positions of 1's are A000040.
Positions of 0's and 2's are A328028.
Positions of terms > 2 are A328189.
Successive pairs of consecutive divisors are counted by A129308.
Cf. A000005, A027750, A033676, A060680, A060681, A060682, A060683, A193829, A328023, A328024, A328194.

f:=func;  g:=func; [g(n):n in [1..100]]; // Marius A. Burtea, Nov 20 2019

Table[Length[Split[Divisors[n],!Divisible[#2,#1]&]]-1,{n,100}]

a(n) = {my(d=divisors(n), nb=0); for (i=2, #d, if ((d[i] % d[i-1]) == 0, nb++)); nb;} \\ Michel Marcus, Oct 05 2019

a(p^k) = k for any prime number p and k >= 0. - Rémy Sigrist, Oct 05 2019

Data section extended up to a(105) by Antti Karttunen, Feb 23 2023

A328161 Numbers n that are prime or whose proper divisors (greater than 1 and less than n) have no consecutive divisibilities.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 24, 25, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 43, 45, 46, 47, 48, 49, 51, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 69, 70, 71, 72, 73, 74, 77, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91

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

Equals the union of A328028 and A000040.
Complement of A328189.
One, primes, and positions of 1's in A328194.
Partitions with no consecutive divisibilities are A328171.
Cf. A000005, A060680, A060775, A067513, A088725, A163870, A328162.

filter:= proc(n) local D,i;
  if isprime(n) then return true 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..100]); # Robert Israel, Oct 11 2019

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

A328188 Number of strict integer partitions of n with all pairs of consecutive parts relatively prime.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 9, 12, 15, 15, 19, 23, 25, 30, 35, 39, 47, 52, 58, 65, 75, 86, 95, 109, 124, 144, 165, 181, 203, 221, 249, 285, 316, 352, 392, 438, 484, 538, 599, 666, 737, 813, 899, 992, 1102, 1215, 1335, 1472, 1621, 1776, 1946, 2137, 2336

Offset: 0

Gus Wiseman, Oct 13 2019

nonn

			The a(1) = 1 through a(15) = 15 partitions (A..F = 10..15):
  1  2  3   4   5   6    7   8    9    A     B     C    D     E     F
        21  31  32  51   43  53   54   73    65    75   76    95    87
                41  321  52  71   72   91    74    B1   85    B3    B4
                         61  431  81   532   83    543  94    D1    D2
                             521  432  541   92    651  A3    653   E1
                                  531  721   A1    732  B2    743   654
                                       4321  731   741  C1    752   753
                                             5321  831  652   761   852
                                                   921  751   851   951
                                                        832   941   A32
                                                        5431  A31   B31
                                                        7321  B21   6531
                                                              5432  7431
                                                              6521  7521
                                                              8321  54321

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The case of compositions is A167606.
The non-strict case is A328172.
The Heinz numbers of these partitions are given by A328335.
Partitions with no pairs of consecutive parts relatively prime are A328187.
Cf. A000837, A018783, A178470, A328028, A328170, A328171, A328187, A328220.

b:= proc(n, i, s) option remember; `if`(i*(i+1)/2 igcd(i, j)=1, s), b(n-i, min(n-i, i-1),
           numtheory[factorset](i)), 0)+b(n, i-1, s)))
    end:
a:= n-> b(n$2, {}):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 13 2019

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MatchQ[#,{_,x_,y_,_}/;GCD[x,y]>1]&]],{n,0,30}]
(* Second program: *)
b[n_, i_, s_] := b[n, i, s] = If[i(i + 1)/2 < n, 0, If[n == 0, 1, If[AllTrue[s,  GCD[i, #] == 1&], b[n - i, Min[n - i, i - 1], FactorInteger[i][[All, 1]]], 0] + b[n, i - 1, s]]];
a[n_] := b[n, n, {}];
a /@ Range[0, 60] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

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

cp's OEIS Frontend

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

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A328172 Number of integer partitions of n with all pairs of consecutive parts relatively prime.

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A328187 Number of integer partitions of n with no pair of consecutive parts relatively prime.

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

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A328026 Number of divisible pairs of consecutive divisors of n.

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A328161 Numbers n that are prime or whose proper divisors (greater than 1 and less than n) have no consecutive divisibilities.

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Maple

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A328188 Number of strict integer partitions of n with all pairs of consecutive parts relatively prime.

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Maple

Mathematica

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

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