A364462 -id:A364462 - cp's OEIS frontend

A365280 a(n) is the least number that starts a run of exactly n numbers that are members of A364462.

12, 324, 6252, 155673, 7445148, 457137900

Offset: 1

Robert Israel, Aug 30 2023

A364462 contains arbitrarily long runs: for example, start with n coprime members t(1),...,t(n) of A365277 and use the Chinese Remainder Theorem to find k such that k == 1-i (mod t(i)) for i = 1 .. n.

			a(1) = 12 = 2^2 * 3 = prime(1) * prime(1) * prime(1+1) is in A364462.
a(2) = 324 = 2^2 * 3^4 is divisible by prime(1) * prime(1) * prime(1+1) and thus in A364462.
a(2) + 1 = 325 = 5^2 * 13 = prime(3) * prime(3) * prime(3+3).
a(3) = 6252 = 2^2 * 3 * 521 is divisible by  prime(1) * prime(1) * prime(1+1).
a(3) + 1 = 6253 = 13^2 * 37 = prime(6) * prime(6) * prime(6+6)
a(3) + 2 = 6254 = 2 * 53 * 49 = prime(1) * prime(16) * prime(17).
a(4) = 155673 = 3^2 * 7^2 * 353 is divisible by prime(2) * prime(2) * prime(2+2).
a(4) + 1 = 155674 = 2 * 277 * 281 = prime(1) * prime(59) * prime(1+59).
a(4) + 2 = 155675 = 5^2 * 13 * 479 is divisible by prime(3) * prime(3) * prime(3+3).
a(4) + 3 = 155676 = 2^2 * 3 * 12973 is divisible by 2 * 2 * 3 = prime(1) * prime(1) * prime(2).
a(5) = 7445148 = 2^2 * 3 * 620429 is divisible by 2 * 2 * 3 = prime(1) * prime(1) * prime(2).
a(5) + 1 = 7445149 = 41^2 * 43 * 103 is divisible by 41 * 43 * 103 = prime(13) * prime(14) * prime(27).
a(5) + 2 = 7445150 = 2 * 5^2 * 17 * 19 * 461 is divisible by 2 * 17 * 19 = prime(1) * prime(7) * prime(8).
a(5) + 3 = 7445151 = 3^2 * 7 * 59 * 2003 is divisible by 3 * 3 * 7 = prime(2) * prime(2) * prime(4).
a(5) + 4 = 7445152 = 2^5 * 11 * 13 * 1627 is divisible by 2 * 11 * 13 = prime(1) * prime(5) * prime(6).

Cf. A364462, A365277.

filter:= proc(n) local F, i,j,m;
  F:= map(t -> `if`(t[2]>=2, numtheory:-pi(t[1])$2, numtheory:-pi(t[1])), ifactors(n)[2]);
  for i from 1 to nops(F)-1 do for j from 1 to i-1 do
    if member(F[i]+F[j],F) then return true fi
  od od;
  false
end proc:
V:= Vector(5): count:= 0: flag:= false:
for x from 1 while count < 5 do
  if filter(x) then
     if not flag then flag:= true; m:= x fi;
  elif flag then
     flag:= false; v:= x-m;
     if V[v] = 0 then count:= count+1; V[v]:= m; fi;
  fi
od:
convert(V,list);

a(6) from David A. Corneth, Sep 01 2023

A237113 Number of partitions of n such that some part is a sum of two other parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 3, 8, 10, 17, 22, 37, 47, 71, 91, 133, 170, 236, 301, 408, 515, 686, 860, 1119, 1401, 1798, 2232, 2829, 3495, 4378, 5381, 6682, 8165, 10060, 12238, 14958, 18116, 22018, 26533, 32071, 38490, 46265, 55318, 66193, 78843, 93949, 111503, 132326

Offset: 0

Clark Kimberling, Feb 04 2014

nonn

These are partitions containing the sum of some 2-element submultiset of the parts, a variation of binary sum-full partitions where parts cannot be re-used, ranked by A364462. The complement is counted by A236912. The non-binary version is A237668. For re-usable parts we have A363225. - Gus Wiseman, Aug 10 2023

			Of the 11 partitions of 6, only these 3 include a part that is a sum of two other parts: [3,2,1], [2,2,1,1], [2,1,1,1,1].  Thus, a(6) = 3.
From _Gus Wiseman_, Aug 09 2023: (Start)
The a(0) = 0 through a(9) = 10 partitions:
  .  .  .  .  (211)  (2111)  (321)    (3211)    (422)      (3321)
                             (2211)   (22111)   (431)      (4221)
                             (21111)  (211111)  (3221)     (4311)
                                                (4211)     (5211)
                                                (22211)    (32211)
                                                (32111)    (42111)
                                                (221111)   (222111)
                                                (2111111)  (321111)
                                                           (2211111)
                                                           (21111111)
(End)

The complement for subsets is A085489, with re-usable parts A007865.
For subsets of {1..n} we have A088809, with re-usable parts A093971.
The complement is counted by A236912, ranks A364461.
The non-binary complement is A237667, ranks A364531.
The non-binary version is A237668, ranks A364532.
With re-usable parts we have A363225, ranks A364348.
The complement with re-usable parts is A364345, ranks A364347.
These partitions have ranks A364462.
The strict case is A364670, with re-usable parts A363226.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A151897, A237984, A325862, A326083, A363260.

z = 20; t = Map[Count[Map[Length[Cases[Map[Total[#] &, Subsets[#, {2}]],  Apply[Alternatives, #]]] &, IntegerPartitions[#]], 0] &, Range[z]] (* A236912 *)
u = PartitionsP[Range[z]] - t  (* A237113, Peter J. C. Moses, Feb 03 2014 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2}]]!={}&]],{n,0,30}] (* Gus Wiseman, Aug 09 2023 *)

a(n) = A000041(n) - A236912(n).

a(0)=0 prepended by Alois P. Heinz, Sep 17 2023

A088809 Number of subsets of {1, ..., n} that are not sum-free.

Original entry on oeis.org

0, 0, 0, 1, 3, 10, 27, 67, 154, 350, 763, 1638, 3450, 7191, 14831, 30398, 61891, 125557, 253841, 511818, 1029863, 2069341, 4153060, 8327646, 16687483, 33422562, 66916342, 133936603, 268026776, 536277032, 1072886163, 2146245056, 4293187682, 8587371116

Offset: 0

Reinhard Zumkeller, Oct 19 2003

nonn

a(n) = 2^n - A085489(n); a non-sum-free subset contains at least one subset {u,v, w} with w=u+v.

A variation of binary sum-full sets where parts cannot be re-used, this sequence counts subsets of {1..n} with an element equal to the sum of two distinct others. The complement is counted by A085489. The non-binary version is A364534. For re-usable parts we have A093971. - Gus Wiseman, Aug 10 2023

			From _Gus Wiseman_, Aug 10 2023: (Start)
The set S = {1,3,6,8} has pair-sums {4,7,9,11,14}, which are all missing from S, so it is not counted under a(8).
The set {1,4,6,7} has pair-sum 1 + 6 = 7, so is counted under a(7).
The a(1) = 0 through a(5) = 10 sets:
  .  .  {1,2,3}  {1,2,3}    {1,2,3}
                 {1,3,4}    {1,3,4}
                 {1,2,3,4}  {1,4,5}
                            {2,3,5}
                            {1,2,3,4}
                            {1,2,3,5}
                            {1,2,4,5}
                            {1,3,4,5}
                            {2,3,4,5}
                            {1,2,3,4,5}
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..75
Eric Weisstein's World of Mathematics, Sum-Free Set
Reinhard Zumkeller, Illustration of initial terms

The complement is counted by A085489, differences A364755.
With re-usable parts we have A093971, for partitions A363225.
The complement for partitions is A236912:
  non-binary A237667,
  ranks A364461,
  strict A364533.
The version for partitions is A237113:
  non-binary A237668,
  ranks A364462,
  strict A364670.
The non-binary version is A364534, complement A151897.
First differences are A364756.
Cf. A000079, A007865, A050291, A051026, A103580, A288728, A326020, A326080, A326083, A364272, A364349.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Total/@Subsets[#,{2}]]!={}&]],{n,0,10}] (* Gus Wiseman, Aug 10 2023 *)

Terms a(32) and beyond from Fausto A. C. Cariboni, Sep 28 2020

A236912 Number of partitions of n such that no part is a sum of two other parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 12, 14, 20, 25, 34, 40, 54, 64, 85, 98, 127, 149, 189, 219, 277, 316, 395, 456, 557, 638, 778, 889, 1070, 1226, 1461, 1667, 1978, 2250, 2645, 3019, 3521, 3997, 4652, 5267, 6093, 6909, 7943, 8982, 10291, 11609, 13251, 14947, 16984, 19104

Offset: 0

Clark Kimberling, Feb 01 2014

nonn

These are partitions containing the sum of no 2-element submultiset of the parts, a variation of binary sum-free partitions where parts cannot be re-used, ranked by A364461. The complement is counted by A237113. The non-binary version is A237667. For re-usable parts we have A364345. - Gus Wiseman, Aug 09 2023

			Of the 11 partitions of 6, only these 3 include a part that is a sum of two other parts: [3,2,1], [2,2,1,1], [2,1,1,1,1].  Thus, a(6) = 11 - 3 = 8.
From _Gus Wiseman_, Aug 09 2023: (Start)
The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (311)    (222)     (322)      (71)
                            (11111)  (411)     (331)      (332)
                                     (3111)    (421)      (521)
                                     (111111)  (511)      (611)
                                               (2221)     (2222)
                                               (4111)     (3311)
                                               (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)
(End)

For subsets of {1..n} we have A085489, complement A088809.
The complement is counted by A237113, ranks A364462.
The non-binary version is A237667, ranks A364531.
The non-binary complement is A237668, ranks A364532.
The version with re-usable parts is A364345, ranks A364347.
The (strict) version for linear combinations of parts is A364350.
These partitions have ranks A364461.
The strict case is A364533, non-binary A364349.
The strict complement is A364670, with re-usable parts A363226.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A007865, A151897, A275972, A325862, A326083, A363225, A363260, A364346, A364755.

z = 20; t = Map[Count[Map[Length[Cases[Map[Total[#] &, Subsets[#, {2}]],  Apply[Alternatives, #]]] &, IntegerPartitions[#]], 0] &, Range[z]] (* A236912 *)
u = PartitionsP[Range[z]] - t  (* A237113, Peter J. C. Moses, Feb 03 2014 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2}]]=={}&]],{n,0,15}] (* Gus Wiseman, Aug 09 2023 *)

a(n) = A000041(n) - A237113(n).

a(0)=1 prepended by Alois P. Heinz, Sep 17 2023

A237667 Number of partitions of n such that no part is a sum of two or more other parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 11, 12, 17, 19, 29, 28, 41, 42, 61, 61, 87, 85, 120, 117, 160, 156, 224, 216, 288, 277, 380, 363, 483, 474, 622, 610, 783, 755, 994, 986, 1235, 1191, 1549, 1483, 1876, 1865, 2306, 2279, 2806, 2732, 3406, 3413, 4091, 4013, 4991, 4895, 5872

Offset: 0

Clark Kimberling, Feb 11 2014

nonn

From Gus Wiseman, Aug 09 2023: (Start)
Includes all knapsack partitions (A108917), but first differs at a(12) = 28, A108917(12) = 25. The difference is accounted for by the non-knapsack partitions: (4332), (5331), (33222).
These are partitions not containing the sum of any non-singleton submultiset of the parts, a variation of non-binary sum-free partitions where parts cannot be re-used, ranked by A364531. The complement is counted by A237668. The binary version is A236912. For re-usable parts we have A364350.
(End)

			For n = 6, the nonqualifiers are 123, 1113, 1122, 11112, leaving a(6) = 7.
From _Gus Wiseman_, Aug 09 2023: (Start)
The partition y = (5,3,1,1) has submultiset (3,1,1) with sum in y, so is not counted under a(10).
The partition y = (5,3,3,1) has no non-singleton submultiset with sum in y, so is counted under a(12).
The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (311)    (222)     (322)      (71)
                            (11111)  (411)     (331)      (332)
                                     (111111)  (421)      (521)
                                               (511)      (611)
                                               (2221)     (2222)
                                               (4111)     (3311)
                                               (1111111)  (5111)
                                                          (11111111)
(End)

Giovanni Resta, Table of n, a(n) for n = 0..100
Giovanni Resta, C program for computing a(0)-a(100)

For subsets of {1..n} we have A151897, binary A085489.
The binary version is A236912, ranks A364461.
The binary complement is A237113, ranks A364462.
The complement is counted by A237668, ranks A364532.
The binary version with re-usable parts is A364345, strict A364346.
The strict case is A364349, binary A364533.
These partitions have ranks A364531.
The complement for subsets is A364534, binary A088809.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A007865, A179009, A325862, A326083, A363225, A363226, A364348, A364350, A364670.

Map[Count[Map[MemberQ[#,Apply[Alternatives,Map[Apply[Plus,#]&, DeleteDuplicates[DeleteCases[Subsets[#],?(Length[#]<2&)]]]]]&, IntegerPartitions[#]],False]&,Range[20]]  (* _Peter J. C. Moses, Feb 10 2014 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2,Length[#]}]]=={}&]],{n,0,15}] (* Gus Wiseman, Aug 09 2023 *)

a(21)-a(53) from Giovanni Resta, Feb 22 2014

A237668 Number of partitions of n such that some part is a sum of two or more other parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 4, 4, 10, 13, 23, 27, 49, 60, 93, 115, 170, 210, 300, 370, 510, 632, 846, 1031, 1359, 1670, 2159, 2630, 3355, 4082, 5130, 6220, 7739, 9360, 11555, 13889, 16991, 20402, 24824, 29636, 35855, 42707, 51309, 60955, 72896, 86328, 102826, 121348

Offset: 0

Clark Kimberling, Feb 11 2014

nonn

These are partitions containing the sum of some non-singleton submultiset of the parts, a variation of non-binary sum-full partitions where parts cannot be re-used, ranked by A364532. The complement is counted by A237667. The binary version is A237113, or A363225 with re-usable parts. This sequence is weakly increasing. - Gus Wiseman, Aug 12 2023

			a(6) = 4 counts these partitions: 123, 1113, 1122, 11112.
From _Gus Wiseman_, Aug 12 2023: (Start)
The a(0) = 0 through a(9) = 13 partitions:
  .  .  .  .  (211)  (2111)  (321)    (3211)    (422)      (3321)
                             (2211)   (22111)   (431)      (4221)
                             (3111)   (31111)   (3221)     (4311)
                             (21111)  (211111)  (4211)     (5211)
                                                (22211)    (32211)
                                                (32111)    (33111)
                                                (41111)    (42111)
                                                (221111)   (222111)
                                                (311111)   (321111)
                                                (2111111)  (411111)
                                                           (2211111)
                                                           (3111111)
                                                           (21111111)
(End)

Giovanni Resta, Table of n, a(n) for n = 0..100
Giovanni Resta, C program for computing a(0)-a(100)

Cf. A179009.
The binary complement is A236912, ranks A364461.
The binary version is A237113, ranks A364462.
The complement is counted by A237667, ranks A364531.
The binary version with re-usable parts is A363225, ranks A364348.
The strict case is A364272.
The binary complement with re-usable parts is A364345, ranks A364347.
These partitions have ranks A364532.
For subsets instead of partitions we have A364534, complement A151897.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A299701 counts distinct subset-sums of prime indices.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A088809, A237984, A325862, A326083, A363226, A364670.

z = 20; m = Map[Count[Map[MemberQ[#, Apply[Alternatives, Map[Apply[Plus, #] &, DeleteDuplicates[DeleteCases[Subsets[#], _?(Length[#] < 2 &)]]]]] &, IntegerPartitions[#]], False] &, Range[z]]; PartitionsP[Range[z]] - m
(* Peter J. C. Moses, Feb 10 2014 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2,Length[#]}]]!={}&]],{n,0,15}] (* Gus Wiseman, Aug 12 2023 *)

a(21)-a(47) from Giovanni Resta, Feb 22 2014

A364347 Numbers k > 0 such that if prime(a) and prime(b) both divide k, then prime(a+b) does not.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85

Offset: 1

Gus Wiseman, Jul 26 2023

nonn

Or numbers without any prime index equal to the sum of two others, allowing re-used parts.

Also Heinz numbers of a type of sum-free partitions counted by A364345.

			We don't have 6 because prime(1), prime(1), and prime(1+1) are all divisors.
The terms together with their prime indices begin:
    1: {}
    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}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}

Subsets of this type are counted by A007865 (sum-free sets).
Partitions of this type are counted by A364345.
The squarefree case is counted by A364346.
The complement is A364348, counted by A363225.
The non-binary version is counted by A364350.
Without re-using parts we have A364461, counted by A236912.
Without re-using parts we have complement A364462, counted by A237113.
A001222 counts prime indices.
A108917 counts knapsack partitions, ranks A299702.
A112798 lists prime indices, sum A056239.
A323092 counts double-free partitions, ranks A320340.
Cf. A093971, A237667, A288728, A325862, A326083, A363226, A364531.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#],Total/@Tuples[prix[#],2]]=={}&]

A364461 Positive integers such that if prime(a)*prime(b) is a divisor, prime(a+b) is not.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76

Offset: 1

Gus Wiseman, Jul 27 2023

nonn

Also Heinz numbers of a type of sum-free partitions not allowing re-used parts, counted by A236912.

			The prime indices of 198 are {1,2,2,5}, which is sum-free even though it is not knapsack (A299702, A299729), so 198 is in the sequence.

Subsets of this type are counted by A085489, with re-usable parts A007865.
Subsets not of this type are counted by A093971, w/ re-usable parts A088809.
Partitions of this type are counted by A236912.
Allowing parts to be re-used gives A364347, counted by A364345.
The complement allowing parts to be re-used is A364348, counted by A363225.
The non-binary version allowing re-used parts is counted by A364350.
The complement is A364462, counted by A237113.
The non-binary version is A364531, counted by A237667, complement A364532.
A001222 counts prime indices.
A108917 counts knapsack partitions, ranks A299702.
A112798 lists prime indices, sum A056239.
Cf. A151897, A288728, A320340, A325862, A325864, A326083, A363226, A364346.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#], Total/@Subsets[prix[#],{2}]]=={}&]

A364348 Numbers with two possibly equal divisors prime(a) and prime(b) such that prime(a+b) is also a divisor.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 70, 72, 78, 84, 90, 96, 102, 105, 108, 114, 120, 126, 130, 132, 133, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252

Offset: 1

Gus Wiseman, Jul 27 2023

nonn

Or numbers with a prime index equal to the sum of two others, allowing re-used parts.

Also Heinz numbers of a type of sum-free partitions counted by A363225.

			We have 6 because prime(1) and prime(1) are both divisors of 6, and prime(1+1) is also.
The terms together with their prime indices begin:
   6: {1,2}
  12: {1,1,2}
  18: {1,2,2}
  21: {2,4}
  24: {1,1,1,2}
  30: {1,2,3}
  36: {1,1,2,2}
  42: {1,2,4}
  48: {1,1,1,1,2}
  54: {1,2,2,2}
  60: {1,1,2,3}
  63: {2,2,4}
  65: {3,6}
  66: {1,2,5}
  70: {1,3,4}
  72: {1,1,1,2,2}

Subsets of this type are counted by A093971, complement A007865.
Partitions of this type are counted by A363225, strict A363226.
The complement is A364347, counted by A364345.
The complement without re-using parts is A364461, counted by A236912.
Without re-using parts we have A364462, counted by A237113.
A001222 counts prime indices.
A108917 counts knapsack partitions, ranks A299702.
A112798 lists prime indices, sum A056239.
A323092 counts double-free partitions, ranks A320340.
Cf. A237667, A275972, A288728, A325862, A326083, A364346.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#],Total/@Tuples[prix[#],2]]!={}&]

A364531 Positive integers with no prime index equal to the sum of prime indices of any nonprime divisor.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, Aug 01 2023

nonn

First differs from A299702 (knapsack) in having 525: {2,3,3,4}.
First differs from A325778 in lacking 462: {1,2,4,5}.
These are the Heinz numbers of partitions whose parts are disjoint from their own non-singleton subset-sums.

Partitions of this type are counted by A237667, strict A364349.
The binary version is A364462, complement A364461.
The complement is A364532, counted by A237668.
A000005 counts divisors, nonprime A033273, composite A055212.
A299701 counts distinct subset-sums of prime indices.
A299702 ranks knapsack partitions, counted by A108917, complement A299729.
A363260 counts partitions disjoint from differences, complement A364467.
Cf. A002865, A236912, A237113, A320340, A326083, A363225, A364347.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]=={}&]

cp's OEIS Frontend

A365280 a(n) is the least number that starts a run of exactly n numbers that are members of A364462.

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A237113 Number of partitions of n such that some part is a sum of two other parts.

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A088809 Number of subsets of {1, ..., n} that are not sum-free.

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A236912 Number of partitions of n such that no part is a sum of two other parts.

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A237667 Number of partitions of n such that no part is a sum of two or more other parts.

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A237668 Number of partitions of n such that some part is a sum of two or more other parts.

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A364347 Numbers k > 0 such that if prime(a) and prime(b) both divide k, then prime(a+b) does not.

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A364461 Positive integers such that if prime(a)*prime(b) is a divisor, prime(a+b) is not.

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