A356736 -id:A356736 - cp's OEIS frontend

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

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

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).

A356734 Heinz numbers of integer partitions with at least one neighborless part.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 78, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Aug 26 2022

nonn

First differs from A319630 in lacking 1 and having 42 (prime indices: {1,2,4}).
A part x is neighborless if neither x - 1 nor x + 1 are parts.
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.

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}

These partitions are counted by A356236.
The singleton case is A356237, counted by A356235 (complement A355393).
The strict case is counted by A356607, complement A356606.
The complement is A356736, counted by A355394.
A001221 counts distinct prime factors, 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).
Cf. A000005, A286470, A287170 (firsts A066205), 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[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]

A356735 Number of distinct parts that have neighbors in the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 3

Offset: 1

Gus Wiseman, Aug 31 2022

nonn

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 the number of distinct prime indices x of n such that either x - 1 or x + 1 is also a prime index of n, where a prime index of n is a number x such that prime(x) divides n.

			The prime indices of 42 are {1,2,4}, of which 1 and 2 have neighbors, so a(42) = 2.
The prime indices of 462 are {1,2,4,5}, all of which have neighbors, so a(462) = 4.
The prime indices of 990 are {1,2,2,3,5}, of which 1, 2, and 3 have neighbors, so a(990) = 3.
The prime indices of 1300 are {1,1,3,3,6}, none of which have neighbors, so a(1300) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

Positions of first appearances are A002110 without 1 (or A231209).
The complement is counted by A356733.
Positions of zeros are A356734.
Positions of positive terms are A356736.
A001221 counts distinct prime factors, sum A001414.
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.
A355393 counts partitions w/o a neighborless singleton, complement A356235.
A355394 counts partitions w/o a neighborless part, complement A356236.
A356226 lists the lengths of maximal gapless submultisets of prime indices:
- length: A287170 (firsts A066205)
- minimum: A356227
- maximum: A356228
- bisected length: A356229
- standard composition: A356230
- Heinz number: A356231
- positions of first appearances: A356232
Cf. A000005, A066312, A286470, A289508, A325160, A328166, A328335, A356233, A356234, A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Union[primeMS[n]], MemberQ[primeMS[n],#-1]|| MemberQ[primeMS[n],#+1]&]],{n,100}]

A356735(n) = if(1==n,0,my(pis=apply(primepi,factor(n)[,1])); omega(n)-sum(i=1, #pis, ((n%prime(pis[i]+1)) && (pis[i]==1 || (n%prime(pis[i]-1)))))); \\ Antti Karttunen, Jan 28 2025

a(n) + A356733(n) = A001221(n).

Data section extended to a(105) by Antti Karttunen, Jan 28 2025

cp's OEIS Frontend

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

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A356734 Heinz numbers of integer partitions with at least one neighborless part.

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A356735 Number of distinct parts that have neighbors in the integer partition with Heinz number n.

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