A007690 -id:A007690 - cp's OEIS frontend

A240085 Number of compositions of n in which no part is unique (every part appears at least twice).

1, 0, 1, 1, 2, 1, 9, 11, 34, 53, 108, 169, 400, 680, 1530, 2984, 6362, 12498, 25766, 50093, 102126, 199309, 400288, 788227, 1581584, 3135117, 6286310, 12532861, 25121292, 50184582, 100627207, 201208477, 403170900, 806534560, 1615151111, 3231224804, 6467909442

Offset: 0

Geoffrey Critzer, Mar 31 2014

nonn

			a(6) = 9 because we have: 3+3, 2+2+2, 2+2+1+1, 2+1+2+1, 2+1+1+2, 1+2+2+1, 1+2+1+2, 1+1+2+2, 1+1+1+1+1+1.

S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Chapman and Hall, 2009.

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..2100 (first 500 terms from Alois P. Heinz)
Vaclav Kotesovec, Graph a(n)/2^n

Cf. A007690.

b:= proc(n, i, t) option remember; `if`(n=0, t!, `if`(i<1, 0,
      b(n, i-1, t) +add(b(n-i*j, i-1, t+j)/j!, j=2..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 31 2014

Table[Length[Level[Map[Permutations,Select[IntegerPartitions[n], Apply[And,Table[Count[#,#[[i]]]>1,{i,1,Length[#]}]]&]],{2}]],{n,0,20}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, t!, If[i < 1, 0, b[n, i - 1, t] + Sum[b[n - i*j, i - 1, t + j]/j!, {j, 2, n/i}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

G.f.: 1 + Sum_{m>1} C(m,x)/(1-x^m) where C(m,x) = x^m + Sum_{i=2..m-2} m! * x^m * C(m-i,x)/(i! * (m-i)! * (1 - x^(m-i))). - John Tyler Rascoe, Jan 09 2025

More terms from Alois P. Heinz, Mar 31 2014

A356607 Number of strict integer partitions of n with at least one neighborless part.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 4, 6, 6, 9, 11, 13, 17, 20, 24, 30, 36, 41, 52, 60, 71, 84, 100, 114, 137, 158, 183, 214, 248, 283, 330, 379, 432, 499, 570, 648, 742, 846, 955, 1092, 1234, 1395, 1580, 1786, 2005, 2270, 2548, 2861, 3216, 3610, 4032, 4526, 5055, 5642, 6304, 7031, 7820, 8720, 9694

Offset: 0

Gus Wiseman, Aug 26 2022

nonn

A part x is neighborless if neither x - 1 nor x + 1 are parts.

			The a(0) = 0 through a(9) = 6 partitions:
  .  (1)  (2)  (3)  (4)   (5)   (6)   (7)    (8)    (9)
                    (31)  (41)  (42)  (52)   (53)   (63)
                                (51)  (61)   (62)   (72)
                                      (421)  (71)   (81)
                                             (431)  (531)
                                             (521)  (621)

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, A356607.py

This is the strict case of A356235 and A356236.
The complement is counted by A356606, non-strict A355393 and A355394.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A073492, A137921, A325160, A328171, A328172, A328187, A328220, A328221.

Table[Length[Select[IntegerPartitions[n],Function[ptn,UnsameQ@@ptn&&Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]

a(31)-a(59) from Lucas A. Brown, Sep 09 2022

A355394 Number of integer partitions of n such that, for all parts x, x - 1 or x + 1 is also a part.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 3, 6, 6, 10, 11, 16, 18, 25, 30, 38, 47, 59, 74, 90, 112, 136, 171, 203, 253, 299, 372, 438, 536, 631, 767, 900, 1085, 1271, 1521, 1774, 2112, 2463, 2910, 3389, 3977, 4627, 5408, 6276, 7304, 8459, 9808, 11338, 13099, 15112, 17404, 20044, 23018, 26450, 30299, 34746, 39711, 45452, 51832

Offset: 0

Gus Wiseman, Aug 26 2022

nonn

These are partitions without a neighborless part, where a part x is neighborless if neither x - 1 nor x + 1 are parts. The first counted partition that does not cover an interval is (5,4,2,1).

			The a(0) = 1 through a(9) = 11 partitions:
  ()  .  .  (21)  (211)  (32)    (321)    (43)      (332)      (54)
                         (221)   (2211)   (322)     (3221)     (432)
                         (2111)  (21111)  (2221)    (22211)    (3222)
                                          (3211)    (32111)    (3321)
                                          (22111)   (221111)   (22221)
                                          (211111)  (2111111)  (32211)
                                                               (222111)
                                                               (321111)
                                                               (2211111)
                                                               (21111111)

Lucas A. Brown, Table of n, a(n) for n = 0..100
Lucas A. Brown, A355394.py

The singleton case is A355393, complement A356235.
The complement is counted by A356236, ranked by A356734.
The strict case is A356606, complement A356607.
These partitions are ranked by A356736.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A066312, A073491, A077855, A328171, A328172, A328187, A328221, A356233, A356237.

Table[Length[Select[IntegerPartitions[n],Function[ptn,!Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]

a(n) = A000041(n) - A356236(n).

a(31)-a(59) from Lucas A. Brown, Sep 04 2022

A055923 Number of partitions of n in which each part occurs a prime number (or 0) times.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 3, 2, 3, 4, 4, 6, 8, 8, 10, 13, 13, 20, 20, 24, 26, 38, 35, 51, 51, 65, 67, 92, 86, 121, 117, 153, 155, 209, 197, 270, 262, 339, 341, 444, 425, 565, 555, 703, 711, 903, 884, 1135, 1128, 1397, 1430, 1766, 1757, 2193, 2214, 2691, 2762, 3344

Offset: 0

Christian G. Bower, Jun 23 2000

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A000041, A007690, A055922.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(isprime(j), b(n-i*j, i-1), 0), j=1..n/i) +b(n, i-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 31 2014

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[PrimeQ[j], b[n-i*j, i-1], 0], {j, 1, n/i}] + b[n, i-1]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

EULER transform of b where b has g.f. Sum {k>0} c(k)*x^k/(1-x^k) where c is inverse EULER transform of characteristic function of prime numbers.

G.f.: Product(1+Sum(x^(i*prime(k)), k=1..infinity), i=1..infinity). - Vladeta Jovovic, Jan 08 2005

A356237 Heinz numbers of integer partitions with a neighborless singleton.

Original entry on oeis.org

2, 3, 5, 7, 10, 11, 13, 14, 17, 19, 20, 21, 22, 23, 26, 28, 29, 31, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93

Offset: 1

Gus Wiseman, Aug 24 2022

nonn

A part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Also numbers that, for some prime index x, are not divisible by prime(x)^2, prime(x - 1), or prime(x + 1). Here, a prime index of n is a number m such that prime(m) divides n.

			The terms together with their prime indices begin:
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  26: {1,6}
  28: {1,1,4}

The complement is counted by A355393.
These partitions are counted by A356235.
Not requiring a singleton gives A356734.
A001221 counts distinct prime factors, with sum A001414.
A003963 multiplies together the prime indices of n.
A007690 counts partitions with no singletons, complement A183558.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A073491 lists numbers with gapless prime indices, complement A073492.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
A356236 counts partitions with a neighborless part, complement A355394.
A356607 counts strict partitions w/ a neighborless part, complement A356606.
Cf. A286470, A289508, A325160, A328166, A328335, A356231, A356233, A356234.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Function[ptn,Or@@Table[Count[ptn,x]==1&&!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]

A355393 Number of integer partitions of n such that, for all parts x of multiplicity 1, either x - 1 or x + 1 is also a part.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 6, 7, 10, 14, 17, 23, 32, 39, 51, 67, 83, 105, 134, 165, 206, 256, 312, 385, 475, 573, 697, 849, 1021, 1231, 1483, 1771, 2121, 2534, 3007, 3575, 4245, 5008, 5914, 6979, 8198, 9626, 11292, 13201, 15430, 18010, 20960, 24389, 28346, 32855, 38066

Offset: 0

Gus Wiseman, Aug 26 2022

nonn

These are partitions without a neighborless singleton, where a part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once.

			The a(0) = 1 through a(8) = 10 partitions:
  ()  .  (11)  (21)   (22)    (32)     (33)      (43)       (44)
               (111)  (211)   (221)    (222)     (322)      (332)
                      (1111)  (2111)   (321)     (2221)     (2222)
                              (11111)  (2211)    (3211)     (3221)
                                       (21111)   (22111)    (3311)
                                       (111111)  (211111)   (22211)
                                                 (1111111)  (32111)
                                                            (221111)
                                                            (2111111)
                                                            (11111111)

This is the singleton case of A355394, complement A356236.
The complement is counted by A356235.
These partitions are ranked by the complement of A356237.
The strict case is A356606, complement A356607.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A073491, A325160, A328171, A328172, A328187, A328220, A356233.

Table[Length[Select[IntegerPartitions[n],Function[ptn,!Or@@Table[Count[ptn,x]==1&&!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]

A356235 Number of integer partitions of n with a neighborless singleton.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 5, 8, 12, 16, 25, 33, 45, 62, 84, 109, 148, 192, 251, 325, 421, 536, 690, 870, 1100, 1385, 1739, 2161, 2697, 3334, 4121, 5071, 6228, 7609, 9303, 11308, 13732, 16629, 20101, 24206, 29140, 34957, 41882, 50060, 59745, 71124, 84598, 100365

Offset: 0

Gus Wiseman, Aug 23 2022

nonn

A part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once. Examples of partitions with a neighborless singleton are: (3), (3,1), (3,1,1), (3,3,1). Examples of partitions without a neighborless singleton are: (3,3,1,1), (4,3,1,1), (3,2,1), (2,1), (3,3).

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)  (3)  (4)   (5)    (6)     (7)      (8)
                 (31)  (41)   (42)    (52)     (53)
                       (311)  (51)    (61)     (62)
                              (411)   (331)    (71)
                              (3111)  (421)    (422)
                                      (511)    (431)
                                      (4111)   (521)
                                      (31111)  (611)
                                               (4211)
                                               (5111)
                                               (41111)
                                               (311111)

The complement is counted by A355393.
This is the singleton case of A356236, complement A355394.
These partitions are ranked by A356237.
The strict case is A356607, complement A356606.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A066205, A325160, A328171, A328172, A328187, A328221, A356233.

Table[Length[Select[IntegerPartitions[n],Min@@Length/@Split[Reverse[#],#1>=#2-1&]==1&]],{n,0,30}]

A356236 Number of integer partitions of n with a neighborless part.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 8, 9, 16, 20, 31, 40, 59, 76, 105, 138, 184, 238, 311, 400, 515, 656, 831, 1052, 1322, 1659, 2064, 2572, 3182, 3934, 4837, 5942, 7264, 8872, 10789, 13109, 15865, 19174, 23105, 27796, 33361, 39956, 47766, 56985, 67871, 80675, 95750, 113416

Offset: 0

Gus Wiseman, Aug 24 2022

nonn

A part x of a partition is neighborless if neither x - 1 nor x + 1 are parts.

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (41)     (33)      (52)
                    (31)    (311)    (42)      (61)
                    (1111)  (11111)  (51)      (331)
                                     (222)     (421)
                                     (411)     (511)
                                     (3111)    (4111)
                                     (111111)  (31111)
                                               (1111111)

The complement is counted by A355394, singleton case A355393.
The singleton case is A356235, ranked by A356237.
The strict case is A356607, complement A356606.
These partitions are ranked by the complement of A356736.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A066205, A112798, A319630, A325160, A328171, A328172, A328187, A328221.

Table[Length[Select[IntegerPartitions[n],Function[ptn,Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]

a(n) = A000041(n) - A355394(n).

A116645 Number of partitions of n having no doubletons. By a doubleton in a partition we mean an occurrence of a part exactly twice (the partition [4,(3,3),2,2,2,(1,1)] of 18 has two doubletons, shown between parentheses).

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 8, 10, 13, 20, 26, 33, 46, 58, 75, 101, 125, 157, 206, 253, 317, 403, 494, 608, 760, 926, 1131, 1393, 1685, 2038, 2487, 2985, 3585, 4331, 5168, 6172, 7392, 8771, 10410, 12382, 14622, 17258, 20400, 23975, 28159, 33115, 38739, 45298, 53000

Offset: 0

Emeric Deutsch and Vladeta Jovovic, Feb 20 2006

nonn

Number of partitions of n having no part that appears exactly twice.

Infinite convolution product of [1,1,0,1,1,1,1,1,1,1] aerated n-1 times. I.e., [1,1,0,1,1,1,1,1,1,1] * [1,0,1,0,0,0,1,0,1,0] * [1,0,0,1,0,0,0,0,0,1] * ... - Mats Granvik, Gary W. Adamson, Aug 07 2009

			a(4) = 3 because we have [4],[3,1] and [1,1,1,1] (the partitions [2,2] and [2,1,1] do not qualify since each of them has a doubleton).

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

Column 0 of A116644.

Cf. A000041, A007690, A116595, A183559, A183568.

h:=product((1-x^(2*j)+x^(3*j))/(1-x^j),j=1..60): hser:=series(h,x=0,60): seq(coeff(hser,x,n),n=0..56);

nn=48;CoefficientList[Series[Product[1/(1-x^i)-x^(2i),{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 30 2013 *)

G.f.: Product_{j>=1} (1-x^(2j)+x^(3j))/(1-x^j).
G.f. for the number of partitions of n having no part that appears exactly m times is Product_{k>0} (1/(1-x^k)-x^(m*k)).
a(n) = A000041(n) - A183559(n) = A183568(n,0) - A183568(n,2). - Alois P. Heinz, Oct 09 2011
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-2*x) + exp(-4*x)) dx = 0.64673501839556449802623523266221107725058748270577037891948... - Vaclav Kotesovec, Jun 12 2025

A118807 Number of partitions of n having no parts with multiplicity 3.

Original entry on oeis.org

1, 1, 2, 2, 5, 6, 9, 12, 19, 24, 34, 43, 62, 77, 105, 132, 177, 220, 287, 356, 462, 570, 723, 888, 1121, 1370, 1705, 2074, 2570, 3111, 3816, 4601, 5617, 6743, 8170, 9777, 11794, 14058, 16858, 20029, 23932, 28334, 33692, 39772, 47133, 55468, 65471, 76840

Offset: 0

Emeric Deutsch, Apr 29 2006

nonn

Column 0 of A118806.

Infinite convolution product of [1,1,1,0,1,1,1,1,1,1] aerated n-1 times. I.e., [1,1,1,0,1,1,1,1,1,1] * [1,0,1,0,1,0,0,0,1,0] * [1,0,0,1,0,0,1,0,0,0] * ... - Mats Granvik, Gary W. Adamson, Aug 07 2009

			a(6) = 9 because among the 11 (=A000041(6)) partitions of 6 only [2,2,2] and [3,1,1,1] have parts with multiplicity 3.

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

Cf. A000041, A118806, A007690, A116645, A183560, A183568.

g:=product(1+x^j+x^(2*j)+x^(4*j)/(1-x^j),j=1..60): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=0..50);

nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k) + x^(4*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)

G.f.: Product_{j>=1} (1 + x^j + x^(2j) + x^(4j)/(1-x^j)).
a(n) = A000041(n) - A183560(n) = A183568(n,0) - A183568(n,3). - Alois P. Heinz, Oct 09 2011
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-3*x) + exp(-5*x)) dx = 0.73597677748514060768682570953508781551028221145343244320009... - Vaclav Kotesovec, Jun 12 2025

cp's OEIS Frontend

A240085 Number of compositions of n in which no part is unique (every part appears at least twice).

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A356607 Number of strict integer partitions of n with at least one neighborless part.

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A355394 Number of integer partitions of n such that, for all parts x, x - 1 or x + 1 is also a part.

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A055923 Number of partitions of n in which each part occurs a prime number (or 0) times.

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A356237 Heinz numbers of integer partitions with a neighborless singleton.

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A355393 Number of integer partitions of n such that, for all parts x of multiplicity 1, either x - 1 or x + 1 is also a part.

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A356235 Number of integer partitions of n with a neighborless singleton.

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A356236 Number of integer partitions of n with a neighborless part.

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