A364462 -id:A364462 - cp's OEIS frontend

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.

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

A367394 Number of integer partitions of n whose length is a semi-sum of the parts.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 3, 6, 7, 14, 15, 25, 30, 46, 54, 80, 97, 139, 169, 229, 282, 382, 461, 607, 746, 962, 1173, 1499, 1817, 2302, 2787, 3467, 4201, 5216, 6260, 7702, 9261, 11294, 13524, 16418, 19572, 23658, 28141, 33756, 40081, 47949, 56662, 67493, 79639

Offset: 0

Gus Wiseman, Nov 19 2023

nonn

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			For the partition y = (3,3,2,1) we have 4 = 3 + 1, so y is counted under a(9).
The a(2) = 1 through a(10) = 14 partitions:
  (11)  .  (211)  (221)  (321)   (421)   (521)    (621)    (721)
                         (2211)  (2221)  (2222)   (3222)   (3322)
                         (3111)  (3211)  (3221)   (3321)   (3331)
                                         (3311)   (4221)   (4222)
                                         (32111)  (4311)   (4321)
                                         (41111)  (32211)  (5221)
                                                  (42111)  (5311)
                                                           (32221)
                                                           (33211)
                                                           (42211)
                                                           (43111)
                                                           (331111)
                                                           (421111)
                                                           (511111)

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
             sum-full  sum-free  comb-full comb-free semi-full semi-free
            -----------------------------------------------------------
partitions:  A367212   A367213   A367218   A367219   A367394*  A367398
strict:      A367214   A367215   A367220   A367221   A367395   A367399
subsets:     A367216   A367217   A367222   A367223   A367396   A367400
ranks:       A367224   A367225   A367226   A367227   A367397   A367401
A000041 counts partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A236912 counts partitions containing no semi-sum, ranks A364461.
A237113 counts partitions containing a semi-sum, ranks A364462.
A237668 counts sum-full partitions, sum-free A237667.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A008284 counts partitions by length, strict A008289.
A365543 counts partitions with a subset-sum k, strict A365661.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A088809, A093971, A126796, A238628, A304792, A363225, A364534, A365541, A365924, A367402.

Table[Length[Select[IntegerPartitions[n], MemberQ[Total/@Subsets[#,{2}], Length[#]]&]], {n,0,10}]

A367395 Number of strict integer partitions of n whose length is the sum of two distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 5, 5, 7, 8, 11, 13, 17, 19, 25, 28, 35, 41, 49, 57, 68, 78, 92, 107, 124, 143, 166, 192, 220, 254, 291, 335, 382, 439, 499, 572, 649, 741, 840, 956, 1080, 1226, 1383, 1566, 1762, 1988, 2235, 2515, 2822, 3166, 3547

Offset: 0

Gus Wiseman, Nov 19 2023

nonn

			The strict partition (5,3,2,1) has 4 = 3 + 1 so is counted under a(11).
The a(6) = 1 through a(17) = 7 strict partitions (A..E = 10..14):
  321  421  521  621  721   821   921   A21   B21   C21    D21    E21
                      4321  5321  6321  5431  6431  6531   7531   7631
                                        7321  8321  7431   8431   8531
                                                    9321   A321   9431
                                                    54321  64321  B321
                                                                  65321
                                                                  74321

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
             sum-full  sum-free  comb-full comb-free semi-full semi-free
            -----------------------------------------------------------
partitions:  A367212   A367213   A367218   A367219   A367394   A367398
strict:      A367214   A367215   A367220   A367221   A367395*  A367399
subsets:     A367216   A367217   A367222   A367223   A367396   A367400
ranks:       A367224   A367225   A367226   A367227   A367397   A367401
A000041 counts partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A088809/A093971 count twofold sum-full subsets.
A236912 counts partitions containing no semi-sum, ranks A364461.
A237113 counts partitions containing a semi-sum, ranks A364462.
A237668 counts sum-full partitions, sum-free A237667.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A008284 counts partitions by length, strict A008289.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A238628, A363225, A364272, A364534, A365661, A365925, A367410, A367411.

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

A367397 Numbers m such that bigomega(m) is the sum of prime indices of some semiprime divisor of m.

Original entry on oeis.org

4, 12, 18, 30, 36, 40, 42, 54, 60, 66, 78, 81, 90, 100, 102, 112, 114, 120, 126, 135, 138, 140, 150, 168, 174, 180, 186, 189, 198, 210, 220, 222, 225, 234, 246, 250, 252, 258, 260, 270, 280, 282, 297, 300, 306, 315, 318, 330, 336, 340, 342, 350, 351, 352, 354

Offset: 1

Gus Wiseman, Nov 21 2023

nonn

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 the partitions counted by A367394.

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
             sum-full  sum-free  comb-full comb-free semi-full semi-free
            -----------------------------------------------------------
partitions:  A367212   A367213   A367218   A367219   A367394   A367398
strict:      A367214   A367215   A367220   A367221   A367395   A367399
subsets:     A367216   A367217   A367222   A367223   A367396   A367400
ranks:       A367224   A367225   A367226   A367227   A367397*  A367401
A325761 ranks partitions whose length is a part, counted by A002865.
A088809 and A093971 count subsets containing semi-sums.
A236912 counts partitions with no semi-sum of the parts, ranks A364461.
A237113 counts partitions with a semi-sum of the parts, ranks A364462.
A304792 counts subset-sums of partitions, strict A365925.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A365381 counts subsets with a subset summing to k, complement A366320.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A229816, A237667, A237668, A238628, A363225, A364272, A365543, A365658, A365918, A366740.

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

A367398 Number of integer partitions of n whose length is not a semi-sum of the parts.

Original entry on oeis.org

1, 1, 1, 3, 4, 6, 8, 12, 16, 23, 28, 41, 52, 71, 89, 122, 151, 200, 246, 321, 398, 510, 620, 794, 968, 1212, 1474, 1837, 2219, 2748, 3302, 4055, 4882, 5942, 7094, 8623, 10275, 12376, 14721, 17661, 20920, 25011, 29516, 35120, 41419, 49053, 57609, 68092, 79780

Offset: 0

Gus Wiseman, Nov 19 2023

nonn

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			For the partition y = (4,3,1) we have semi-sums {4,5,7}, which do not include 3 (the length of y), so y is counted under a(8).
The a(1) = 1 through a(8) = 16 partitions:
  (1)  (2)  (3)    (4)     (5)      (6)       (7)        (8)
            (21)   (22)    (32)     (33)      (43)       (44)
            (111)  (31)    (41)     (42)      (52)       (53)
                   (1111)  (311)    (51)      (61)       (62)
                           (2111)   (222)     (322)      (71)
                           (11111)  (411)     (331)      (332)
                                    (21111)   (511)      (422)
                                    (111111)  (4111)     (431)
                                              (22111)    (611)
                                              (31111)    (4211)
                                              (211111)   (5111)
                                              (1111111)  (22211)
                                                         (221111)
                                                         (311111)
                                                         (2111111)
                                                         (11111111)

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
             sum-full  sum-free  comb-full comb-free semi-full semi-free
            -----------------------------------------------------------
partitions:  A367212   A367213   A367218   A367219   A367394   A367398*
strict:      A367214   A367215   A367220   A367221   A367395   A367399
subsets:     A367216   A367217   A367222   A367223   A367396   A367400
ranks:       A367224   A367225   A367226   A367227   A367397   A367401
A000041 counts partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A236912 counts partitions containing no semi-sum, ranks A364461.
A237113 counts partitions containing a semi-sum, ranks A364462.
A237667 counts sum-free partitions, sum-full A237668.
A366738 counts semi-sums of partitions, strict A366741.
A367402 counts partitions with covering semi-sums, complement A367403.
Triangles:
A008284 counts partitions by length, strict A008289.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A126796, A304792, A363225, A364272, A365543.

Table[Length[Select[IntegerPartitions[n],FreeQ[Total/@Subsets[#,{2}],Length[#]]&]],{n,0,10}]

A367401 Numbers m such that bigomega(m) is not the sum of prime indices of any semiprime divisor of m.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, Nov 21 2023

nonn

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 the partitions counted by A367398.

			60 has semiprime divisor 10 with prime indices {1,3} summing to 4 = bigomega(60), so 60 is not in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   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}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
             sum-full  sum-free  comb-full comb-free semi-full semi-free
            -----------------------------------------------------------
partitions:  A367212   A367213   A367218   A367219   A367394   A367398
strict:      A367214   A367215   A367220   A367221   A367395   A367399
subsets:     A367216   A367217   A367222   A367223   A367396   A367400
ranks:       A367224   A367225   A367226   A367227   A367397   A367401*
A002865 counts partitions w/ length, complement A229816, ranks A325761.
A088809 and A093971 count subsets containing semi-sums.
A236912 counts partitions with no semi-sum of the parts, ranks A364461.
A237113 counts partitions with a semi-sum of the parts, ranks A364462.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A365381 counts subsets with a subset summing to k, complement A366320.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A237667, A238628, A304792, A363225, A364272, A365543, A365658, A365918, A366740.

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

A365277 Numbers of the form prime(i)prime(j)prime(i+j).

Original entry on oeis.org

12, 30, 63, 70, 154, 165, 273, 286, 325, 442, 561, 595, 646, 741, 874, 931, 1045, 1173, 1334, 1495, 1653, 1771, 1798, 2139, 2294, 2465, 2639, 2945, 3034, 3219, 3509, 3526, 3689, 3813, 4042, 4255, 4433, 4773, 4921, 4982, 5781, 5945, 6253, 6254, 6601, 6665, 6837, 6919, 7198, 8174, 8319, 8569, 8695

Offset: 1

Robert Israel, Aug 30 2023

nonn

Members of A364462 that have no proper divisor in A364462.

			a(3) = 63 is a term because 63 = 3^2 * 7 = prime(2) * prime(2) * prime(2+2).

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A014612 and A364462.

N:= 10^4: # for terms <= N
S:= NULL:
for i from 1 do
  p:= ithprime(i);
  if 2*p^2 > N then break fi;
  for j from 1 to i do
    v:= p * ithprime(j)*ithprime(i+j);
    if v > N then break fi;
    S:= S,v
  od
od:
sort([S]);

A364434 a(1) = 12; for n >= 2, a(n) = least positive integer of the form prime(m)prime(n-m)prime(n) with m >= 1.

Original entry on oeis.org

12, 12, 30, 63, 154, 273, 442, 646, 874, 1334, 1798, 2294, 3034, 3526, 4042, 4982, 6254, 7198, 8174, 9514, 10366, 11534, 13114, 14774, 17266, 19594, 20806, 22042, 23326, 24634, 28702, 33274, 35894, 38086, 41422, 44998, 47414, 51182, 54442, 57782, 61934, 64798, 69142

Offset: 1

David A. Corneth, Aug 31 2023

nonn

Also a(n) is the least positive integer in A364462 that is divisible by prime(n).
This sequence is strictly increasing for n > 1.
Proof by contradiction:
Suppose a(n) >= a(n+1) = prime(n + 1) * prime(m) * prime(n + 1 - m) for some 1 <= m < n + 1. Then, as prime(n + 1) > prime(n) and prime(n + 1 - m) > prime(n - m) we have a(n) >= a(n+1) = prime(n + 1) * prime(m) * prime(n + 1 - m) > prime(n) * prime(m) * prime(n - m) >= a(n). A contradiction.
We contradicted a(n) >= a(n + 1) for n > 1. Therefore for n > 1 we have a(n) < a(n + 1). a(1) = a(2) because prime(0) does not exist.
This sequence could help in finding terms for A365280. Once an upper bound is chosen for a search, one could find the largest prime factor that could part of the product prime(m)*prime(t)*prime(m+t) <= u. This way for any prime p > prime(m+t) we do not need to compute primepi(p) saving a bunch of time in checking if a term is in A364462.

			For n = 2, we take m=1 and get a(2) = prime(1)*prime(1)*prime(2) = 12.

Cf. A364462, A365280.

first(n) = {my(pr = primes(n), res = vector(n, i, oo)); res[1] = 12; for(i = 2, n, for(j = 1, i\2, res[i] = min(res[i], pr[j]*pr[i-j])); res[i]*=pr[i]); res} \\ David A. Corneth, Aug 31 2023

cp's OEIS Frontend

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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Mathematica

A367394 Number of integer partitions of n whose length is a semi-sum of the parts.

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A367395 Number of strict integer partitions of n whose length is the sum of two distinct parts.

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A367397 Numbers m such that bigomega(m) is the sum of prime indices of some semiprime divisor of m.

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A367398 Number of integer partitions of n whose length is not a semi-sum of the parts.

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Mathematica

A367401 Numbers m such that bigomega(m) is not the sum of prime indices of any semiprime divisor of m.

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Mathematica

A365277 Numbers of the form prime(i)*prime(j)*prime(i+j).

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Maple

A364434 a(1) = 12; for n >= 2, a(n) = least positive integer of the form prime(m)*prime(n-m)*prime(n) with m >= 1.

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PARI

A365277 Numbers of the form prime(i)prime(j)prime(i+j).

A364434 a(1) = 12; for n >= 2, a(n) = least positive integer of the form prime(m)prime(n-m)prime(n) with m >= 1.