A237113 -id:A237113 - cp's OEIS frontend

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

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]]!={}&]

A364533 Number of strict integer partitions of n containing the sum of no pair of distinct parts. A variation of sum-free strict partitions.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 11, 11, 15, 15, 21, 22, 28, 32, 38, 40, 51, 55, 65, 74, 83, 94, 111, 119, 136, 160, 174, 196, 222, 252, 273, 315, 341, 391, 425, 477, 518, 602, 636, 719, 782, 886, 944, 1073, 1140, 1302, 1380, 1553, 1651, 1888, 1995, 2224, 2370

Offset: 0

Gus Wiseman, Aug 02 2023

nonn

			The a(1) = 1 through a(12) = 11 partitions (A..C = 10..12):
  1   2   3    4    5    6    7     8     9     A     B     C
          21   31   32   42   43    53    54    64    65    75
                    41   51   52    62    63    73    74    84
                              61    71    72    82    83    93
                              421   521   81    91    92    A2
                                          432   631   A1    B1
                                          531   721   542   543
                                          621         632   732
                                                      641   741
                                                      731   831
                                                      821   921

For subsets of {1..n} we have A085489, complement A088809.
The non-strict version is A236912, complement A237113, ranked by A364461.
Allowing re-used parts gives A364346.
The non-binary version is A364349, non-strict A237667 (complement A237668).
The linear combination-free version is A364350.
The complement in strict partitions is A364670, w/ re-used parts A363226.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, strict A275972.
A151897 counts sum-free subsets, complement A364534.
A323092 counts double-free partitions, ranks A320340.
Cf. A007865, A025065, A240861, A325862, A326083, A364272, A364345, A364347.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Intersection[#, Total/@Subsets[#,{2}]] == {}&]],{n,0,30}]

A365311 Number of strict integer partitions with sum <= n that can be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 11, 12, 20, 24, 35, 38, 63, 63, 92, 112, 148, 160, 230, 244, 339, 383, 478, 533, 726, 781, 978, 1123, 1394, 1526, 1960, 2112, 2630, 2945, 3518, 3964, 4856, 5261, 6307, 7099, 8464, 9258, 11140, 12155, 14419, 16093, 18589, 20565, 24342, 26597, 30948

Offset: 0

Gus Wiseman, Sep 04 2023

nonn

			The strict partition (6,3) cannot be linearly combined to obtain 10, so is not counted under a(10).
The strict partition (4,2) has 6 = 1*4 + 1*2 so is counted under a(6), but (4,2) cannot be linearly combined to obtain 7 so is not counted under a(7).
The a(1) = 1 through a(7) = 12 strict partitions:
  (1)  (1)  (1)    (1)    (1)    (1)      (1)
       (2)  (3)    (2)    (5)    (2)      (7)
            (2,1)  (4)    (2,1)  (3)      (2,1)
                   (2,1)  (3,1)  (6)      (3,1)
                   (3,1)  (3,2)  (2,1)    (3,2)
                          (4,1)  (3,1)    (4,1)
                                 (3,2)    (4,3)
                                 (4,1)    (5,1)
                                 (4,2)    (5,2)
                                 (5,1)    (6,1)
                                 (3,2,1)  (3,2,1)
                                          (4,2,1)

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

For positive coefficients we have A088314.
The positive complement is counted by A088528.
The version for subsets is A365073.
The complement is counted by A365312.
For non-strict partitions we have A365379.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A093971, A237113, A237668, A326080, A363225, A364272, A364534, A364914, A365043, A365314, A365320.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Select[Join@@Array[IntegerPartitions,n],UnsameQ@@#&],combs[n,#]!={}&]],{n,10}]

from math import isqrt
from sympy.utilities.iterables import partitions
def A365311(n):
    a = {tuple(sorted(set(p))) for p in partitions(n)}
    return sum(1 for m in range(1,n+1) for b in partitions(m,m=isqrt(1+(n<<3))>>1) if max(b.values()) == 1 and any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023

a(26)-a(50) from Chai Wah Wu, Sep 13 2023

A365377 Number of subsets of {1..n} without a subset summing to n.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 17, 26, 49, 72, 134, 201, 366, 544, 984, 1436, 2614, 3838, 6770, 10019, 17767, 25808, 45597, 66671, 116461, 169747, 295922, 428090, 750343, 1086245, 1863608, 2721509, 4705456, 6759500, 11660244, 16877655, 28879255, 41778027, 71384579, 102527811, 176151979

Offset: 0

Gus Wiseman, Sep 08 2023

nonn

			The a(1) = 1 through a(6) = 17 subsets:
  {}  {}   {}   {}     {}     {}
      {1}  {1}  {1}    {1}    {1}
           {2}  {2}    {2}    {2}
                {3}    {3}    {3}
                {1,2}  {4}    {4}
                {2,3}  {1,2}  {5}
                       {1,3}  {1,2}
                       {2,4}  {1,3}
                       {3,4}  {1,4}
                              {2,3}
                              {2,5}
                              {3,4}
                              {3,5}
                              {4,5}
                              {1,3,4}
                              {2,3,5}
                              {3,4,5}

David A. Corneth, Table of n, a(n) for n = 0..60

The complement w/ re-usable parts is A365073.
The complement is counted by A365376.
The version with re-usable parts is A365380.
A000009 counts sets summing to n, multisets A000041.
A000124 counts distinct possible sums of subsets of {1..n}.
A124506 appears to count combination-free subsets, differences of A326083.
A364350 counts combination-free strict partitions, complement A364839.
A365046 counts combination-full subsets, differences of A364914.
A365381 counts subsets of {1..n} with a subset summing to k.
Cf. A007865, A085489, A088809, A093971, A103580, A151897, A236912, A237113, A237668, A326080, A364349, A364534.

Table[Length[Select[Subsets[Range[n]], FreeQ[Total/@Subsets[#],n]&]],{n,0,10}]

isok(s, n) = forsubset(#s, ss, if (vecsum(vector(#ss, k, s[ss[k]])) == n, return(0))); return(1);
a(n) = my(nb=0); forsubset(n, s, if (isok(s, n), nb++)); nb; \\ Michel Marcus, Sep 09 2023

from itertools import combinations, chain
from sympy.utilities.iterables import partitions
def A365377(n):
    if n == 0: return 0
    nset = set(range(1,n+1))
    s, c = [set(p) for p in partitions(n,m=n,k=n) if max(p.values(),default=1) == 1], 1
    for a in chain.from_iterable(combinations(nset,m) for m in range(2,n+1)):
        if sum(a) >= n:
            aset = set(a)
            for p in s:
                if p.issubset(aset):
                    c += 1
                    break
    return (1<Chai Wah Wu, Sep 09 2023

a(n) = 2^n-A365376(n). - Chai Wah Wu, Sep 09 2023

a(16)-a(27) from Michel Marcus, Sep 09 2023
a(28)-a(32) from Chai Wah Wu, Sep 09 2023
a(33)-a(35) from Chai Wah Wu, Sep 10 2023
More terms from David A. Corneth, Sep 10 2023

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[#]]}]]=={}&]

A364532 Positive integers with a prime index equal to the sum of prime indices of some nonprime divisor. Heinz numbers of a variation of sum-full partitions.

Original entry on oeis.org

12, 24, 30, 36, 40, 48, 60, 63, 70, 72, 80, 84, 90, 96, 108, 112, 120, 126, 132, 140, 144, 150, 154, 156, 160, 165, 168, 180, 189, 192, 198, 200, 204, 210, 216, 220, 224, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 320, 324, 325, 330

Offset: 1

Gus Wiseman, Aug 01 2023

nonn

First differs from A299729 (non-knapsack) in lacking 525: {2,3,3,4}.
First differs from A325777 in having 462: {1,2,4,5} and lacking 675:{2,2,2,3,3}.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
These are the Heinz numbers of partitions containing the sum of some non-singleton submultiset.

			The terms together with their prime indices begin:
  12: {1,1,2}
  24: {1,1,1,2}
  30: {1,2,3}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  60: {1,1,2,3}
  63: {2,2,4}
  70: {1,3,4}
  72: {1,1,1,2,2}
  80: {1,1,1,1,3}
  84: {1,1,2,4}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}

Partitions not of this type are counted by A237667, strict A364349.
Partitions of this type are counted by A237668, strict A364272.
The binary complement is A364461, re-usable A364347 (counted by A364345).
The binary version is A364462, re-usable A364348 (counted by A363225).
The complement is A364531.
Subsets of this type are counted by A364534, complement A151897.
A000005 counts divisors, nonprime A033273, composite A055212.
A001222 counts prime indices.
A108917 counts knapsack partitions, strict A275972, for subsets A325864.
A112798 lists prime indices, sum A056239.
A299701 counts distinct subset-sums of prime indices.
A299702 ranks knapsack partitions, complement A299729.
Cf. A085489, A088809, A093971, A236912, A237113, A301900, A326083, A364350.

Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]!={}&]

A364463 Number of subsets of {1..n} with elements disjoint from first differences of elements.

Original entry on oeis.org

1, 2, 3, 6, 10, 18, 30, 54, 92, 167, 290, 525, 935, 1704, 3082, 5664, 10386, 19249, 35701, 66702, 124855, 234969, 443174, 839254, 1592925, 3032757, 5786153, 11066413, 21204855, 40712426, 78294085, 150815154, 290922900, 561968268, 1086879052, 2104570243

Offset: 0

Gus Wiseman, Jul 27 2023

nonn

In other words, no element is the difference of two consecutive elements.
From David A. Corneth, Aug 02 2023: (Start)
As subsets counted in a(n) are also counted in a(n+1) and {n+1} is a subset counted in a(n+1) but not a(n), a(n + 1) > a(n) for n >= 1.
As every subset counted in a(n + 1) that contains n+1 can be found from some subset counted in a(n) by appending n+1 and every subset counted in a(n) not containing n + 1 is counted in a(n + 1), a(n+1) <= 2*a(n). (End)

			The a(0) = 1 through a(5) = 18 subsets:
  {}  {}   {}   {}     {}       {}
      {1}  {1}  {1}    {1}      {1}
           {2}  {2}    {2}      {2}
                {3}    {3}      {3}
                {1,3}  {4}      {4}
                {2,3}  {1,3}    {5}
                       {1,4}    {1,3}
                       {2,3}    {1,4}
                       {3,4}    {1,5}
                       {2,3,4}  {2,3}
                                {2,5}
                                {3,4}
                                {3,5}
                                {4,5}
                                {1,3,5}
                                {2,3,4}
                                {3,4,5}
                                {2,3,4,5}

For all differences of pairs of elements we have A007865.
For partitions instead of subsets we have A363260, strict A364464.
The complement is counted by A364466.
A000041 counts integer partitions, strict A000009.
A364465 counts subsets with distinct first differences, partitions A325325.
Cf. A011782, A025065, A229816, A236912, A237113, A237667, A240861, A320347, A323092, A326083, A364347.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Differences[#]]=={}&]],{n,0,10}]

from itertools import combinations
def A364463(n): return sum(1 for l in range(n+1) for c in combinations(range(1,n+1),l) if set(c).isdisjoint({c[i+1]-c[i] for i in range(l-1)})) # Chai Wah Wu, Sep 26 2023

a(n) < a(n + 1) <= 2 * a(n). - David A. Corneth, Aug 02 2023

a(21)-a(29) from David A. Corneth, Aug 02 2023
a(30)-a(32) from Chai Wah Wu, Sep 26 2023
a(33)-a(35) from Chai Wah Wu, Sep 27 2023

A364912 Triangle read by rows where T(n,k) is the number of ways to write n as a positive linear combination of an integer partition of k.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 4, 4, 5, 0, 1, 4, 8, 7, 7, 0, 1, 6, 13, 17, 12, 11, 0, 1, 6, 18, 28, 30, 19, 15, 0, 1, 8, 24, 50, 58, 53, 30, 22

Offset: 0

Gus Wiseman, Aug 20 2023

A way of writing n as a positive linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i > 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(2,2)) are a way of writing 8 as a positive linear combination of (1,1,2), namely 8 = 3*1 + 1*1 + 2*2.

			Triangle begins:
  1
  0  1
  0  1  2
  0  1  2  3
  0  1  4  4  5
  0  1  4  8  7  7
  0  1  6 13 17 12 11
  0  1  6 18 28 30 19 15
  0  1  8 24 50 58 53 30 22
Row n = 4 counts the following linear combinations:
  .  1*4  2*2      2*1+1*2      4*1
          1*1+1*3  1*1+1*1+1*2  3*1+1*1
          1*2+1*2  1*1+1*2+1*1  2*1+2*1
          1*3+1*1  1*2+1*1+1*1  2*1+1*1+1*1
                                1*1+1*1+1*1+1*1
Row n = 5 counts the following linear combinations:
  .  1*5  1*1+1*4  2*1+1*3      3*1+1*2          5*1
          1*2+1*3  2*2+1*1      2*1+1*1+1*2      4*1+1*1
          1*3+1*2  1*1+1*1+1*3  2*1+1*2+1*1      3*1+2*1
          1*4+1*1  1*1+1*2+1*2  1*1+1*1+1*1+1*2  3*1+1*1+1*1
                   1*1+1*3+1*1  1*1+1*1+1*2+1*1  2*1+2*1+1*1
                   1*2+1*1+1*2  1*1+1*2+1*1+1*1  2*1+1*1+1*1+1*1
                   1*2+1*2+1*1  1*2+1*1+1*1+1*1  1*1+1*1+1*1+1*1+1*1
                   1*3+1*1+1*1
Array begins:
  1   0   0   0    0    0    0     0
  1   1   1   1    1    1    1     1
  2   2   4   4    6    6    8     8
  3   4   8   13   18   24   33    40
  5   7   17  28   50   70   107   143
  7   12  30  58   108  179  286   428
  11  19  53  109  223  394  696   1108
  15  30  86  194  420  812  1512  2619

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

Row k = 0 is A000007.
Row k = 1 is A000012.
Column n = 0 is A000041.
Column n = 1 is A000070.
Row sums are A006951.
Row k = 2 is A052928 except initial terms.
The case of strict integer partitions is A116861.
Central column is T(2n,n) = A(n,n) = A364907(n).
With rows reversed we have the nonnegative version A365004.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A237113, A364272, A364910, A364911, A364914, A364915, A365002.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combp[n,ptn],{ptn,IntegerPartitions[k]}]],{n,0,6},{k,0,n}]
- or -
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combs[n-k,ptn],{ptn,IntegerPartitions[k]}]],{n,0,6},{k,0,n}]

As an array, also the number of ways to write n-k as a nonnegative linear combination of an integer partition of k (see programs).

A364670 Number of strict integer partitions of n with a part equal to the sum of two distinct others. A variation of sum-full strict partitions.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 1, 4, 3, 7, 6, 10, 10, 14, 16, 24, 25, 34, 39, 48, 59, 71, 81, 103, 120, 136, 166, 194, 226, 260, 312, 353, 419, 473, 557, 636, 742, 824, 974, 1097, 1266, 1418, 1646, 1837, 2124, 2356, 2717, 3029, 3469, 3830, 4383, 4884, 5547

Offset: 0

Gus Wiseman, Aug 03 2023

nonn

			The a(6) = 1 through a(16) = 10 strict partitions (A = 10):
  321  .  431  .  532   5321  642   5431  743   6432   853
                  541         651   6421  752   6531   862
                  4321        5421  7321  761   7431   871
                              6321        5432  7521   6532
                                          6431  9321   6541
                                          6521  54321  7432
                                          8321         7621
                                                       8431
                                                       A321
                                                       64321

For subsets of {1..n} we have A088809, complement A085489.
The non-strict version is A237113, complement A236912.
The non-binary complement is A237667, ranks A364532.
Allowing re-used parts gives A363226, non-strict A363225.
The non-binary version is A364272, non-strict A237668.
The complement is A364533, non-binary A364349.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, strict A275972, ranks A299702.
A323092 counts double-free partitions, ranks A320340.
Cf. A007865, A025065, A093971, A111133, A151897, A240861, A325862, A364346, A364350, A364534.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Intersection[#, Total/@Subsets[#,{2}]]!={}&]],{n,0,30}]

A364466 Number of subsets of {1..n} where some element is a difference of two consecutive elements.

Original entry on oeis.org

0, 0, 1, 2, 6, 14, 34, 74, 164, 345, 734, 1523, 3161, 6488, 13302, 27104, 55150, 111823, 226443, 457586, 923721, 1862183, 3751130, 7549354, 15184291, 30521675, 61322711, 123151315, 247230601, 496158486, 995447739, 1996668494, 4004044396, 8027966324, 16092990132, 32255168125

Offset: 0

Gus Wiseman, Jul 31 2023

nonn

In other words, the elements are not disjoint from their own first differences.

			The a(0) = 0 through a(5) = 14 subsets:
  .  .  {1,2}  {1,2}    {1,2}      {1,2}
               {1,2,3}  {2,4}      {2,4}
                        {1,2,3}    {1,2,3}
                        {1,2,4}    {1,2,4}
                        {1,3,4}    {1,2,5}
                        {1,2,3,4}  {1,3,4}
                                   {1,4,5}
                                   {2,3,5}
                                   {2,4,5}
                                   {1,2,3,4}
                                   {1,2,3,5}
                                   {1,2,4,5}
                                   {1,3,4,5}
                                   {1,2,3,4,5}

For differences of all pairs we have A093971, complement A196723.
For partitions we have A363260, complement A364467.
The complement is counted by A364463.
For subset-sums instead of differences we have A364534, complement A325864.
For strict partitions we have A364536, complement A364464.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A050291 counts double-free subsets, complement A088808.
A108917 counts knapsack partitions, strict A275972.
A325325 counts partitions with all distinct differences, strict A320347.
Cf. A011782, A212986, A237113, A325877, A325878, A326083, A363225.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Differences[#]]!={}&]],{n,0,10}]

from itertools import combinations
def A364466(n): return sum(1 for l in range(n+1) for c in combinations(range(1,n+1),l) if not set(c).isdisjoint({c[i+1]-c[i] for i in range(l-1)})) # Chai Wah Wu, Sep 26 2023

a(n) = 2^n - A364463(n). - Chai Wah Wu, Sep 26 2023

a(21)-a(32) from Chai Wah Wu, Sep 26 2023

a(33)-a(35) from Chai Wah Wu, Sep 27 2023

cp's OEIS Frontend

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

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A364533 Number of strict integer partitions of n containing the sum of no pair of distinct parts. A variation of sum-free strict partitions.

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A365311 Number of strict integer partitions with sum <= n that can be linearly combined using nonnegative coefficients to obtain n.

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A365377 Number of subsets of {1..n} without a subset summing to n.

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

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A364532 Positive integers with a prime index equal to the sum of prime indices of some nonprime divisor. Heinz numbers of a variation of sum-full partitions.

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A364463 Number of subsets of {1..n} with elements disjoint from first differences of elements.

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A364912 Triangle read by rows where T(n,k) is the number of ways to write n as a positive linear combination of an integer partition of k.

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