A353399 -id:A353399 - cp's OEIS frontend

A353396 Number of integer partitions of n whose Heinz number has prime shadow equal to the product of prime shadows of its parts.

1, 0, 1, 1, 0, 2, 0, 3, 1, 3, 4, 3, 7, 5, 9, 8, 12, 15, 15, 20, 21, 25, 31, 33, 38, 42, 46, 56, 61, 67, 78, 76, 96, 100, 114, 131, 130, 157, 157, 185, 200, 214, 236, 253, 275, 302, 333, 351, 386, 408, 440, 486, 515, 564, 596, 633, 691, 734, 800, 854, 899, 964

Offset: 0

Gus Wiseman, May 15 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.

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			The a(8) = 1 through a(14) = 9 partitions (A..D = 10..13):
  (53)  (72)    (73)    (B)     (75)     (D)      (B3)
        (621)   (532)   (A1)    (651)    (B2)     (752)
        (4221)  (631)   (4331)  (732)    (A21)    (761)
                (4411)          (6321)   (43321)  (A31)
                                (6411)   (44311)  (C11)
                                (43221)           (6521)
                                (44211)           (9221)
                                                  (54221)
                                                  (64211)

The LHS (prime shadow) is A181819, with an inverse A181821.
The RHS (product of prime shadows) is A353394, first appearances A353397.
These partitions are ranked by A353395.
A related comparison is A353398, ranked by A353399.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914, product A005361.
A239455 counts Look-and-Say partitions, ranked by A351294.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000005, A002033, A143773, A182850, A316428, A325131, A325702, A325755, A353389, A353426.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Table[Length[Select[IntegerPartitions[n],Times@@red/@#==red[Times@@Prime/@#]&]],{n,0,15}]

A353504 Number of integer partitions of n whose product is less than the product of their multiplicities.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 4, 4, 6, 6, 8, 9, 11, 12, 14, 14, 18, 21, 23, 26, 29, 29, 33, 36, 39, 40, 43, 44, 50, 53, 55, 59, 65, 69, 72, 78, 79, 81, 85, 92, 95, 97, 100, 103, 108, 109, 112, 118, 124, 129, 137, 139, 142, 149, 155, 159, 165, 166, 173, 178, 181, 187

Offset: 0

Gus Wiseman, May 19 2022

nonn

			The a(2) = 1 through a(9) = 6 partitions:
  11   111   1111   2111    21111    22111     221111     222111
                    11111   111111   31111     311111     411111
                                     211111    2111111    2211111
                                     1111111   11111111   3111111
                                                          21111111
                                                          111111111

LHS (product of parts) is counted by A339095, ranked by A003963.
RHS (product of multiplicities) is counted by A266477, ranked by A005361.
The version for greater instead of less is A353505.
The version for equal instead of less is A353506, ranked by A353503.
A181819 gives prime shadow, with an inverse A181821.
A353398 counts partitions with the same product of multiplicities as of shadows, ranked by A353399.
Cf. A002033, A008284, A085629, A097318, A098859, A114640, A116608, A118914, A124010, A304678, A353394, A353500, A353507.

Table[Length[Select[IntegerPartitions[n],Times@@#

A353505 Number of integer partitions of n whose product is greater than the product of their multiplicities.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 7, 11, 17, 24, 35, 47, 66, 89, 121, 162, 214, 276, 362, 464, 599, 763, 971, 1219, 1537, 1918, 2393, 2966, 3668, 4512, 5549, 6784, 8287, 10076, 12238, 14807, 17898, 21556, 25931, 31094, 37243, 44486, 53075, 63158, 75069, 89025, 105447, 124636

Offset: 0

Gus Wiseman, May 19 2022

nonn

			The a(0) = 0 through a(7) = 11 partitions:
  .  .  (2)  (3)   (4)   (5)    (6)    (7)
             (21)  (22)  (32)   (33)   (43)
                   (31)  (41)   (42)   (52)
                         (221)  (51)   (61)
                         (311)  (222)  (322)
                                (321)  (331)
                                (411)  (421)
                                       (511)
                                       (2221)
                                       (3211)
                                       (4111)

RHS (product of multiplicities) is counted by A266477, ranked by A005361.
LHS (product of parts) is counted by A339095, ranked by A003963.
The version for less instead of greater is A353504.
The version for equality is A353506, ranked by A353503.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A353398 counts partitions with the same products of multiplicities as of shadows, ranked by A353399.
Cf. A002033, A008284, A008619, A085629, A097318, A098859, A114640, A116608, A130091, A304678, A353394, A353507.

Table[Length[Select[IntegerPartitions[n],Times@@#>Times@@Length/@Split[#]&]],{n,0,30}]

A353699 Heinz numbers of integer partitions whose product equals their length.

Original entry on oeis.org

2, 6, 20, 36, 56, 176, 240, 416, 864, 1088, 1344, 2432, 3200, 5888, 8448, 14848, 23040, 31744, 35840, 39936, 75776, 167936, 208896, 331776, 352256, 450560, 516096, 770048, 802816, 933888, 1736704, 2457600, 3866624, 4259840, 4521984, 7995392, 12976128, 17563648

Offset: 1

Gus Wiseman, May 19 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.

			The terms together with their prime indices begin:
      2: {1}
      6: {1,2}
     20: {1,1,3}
     36: {1,1,2,2}
     56: {1,1,1,4}
    176: {1,1,1,1,5}
    240: {1,1,1,1,2,3}
    416: {1,1,1,1,1,6}
    864: {1,1,1,1,1,2,2,2}
   1088: {1,1,1,1,1,1,7}
   1344: {1,1,1,1,1,1,2,4}
   2432: {1,1,1,1,1,1,1,8}
   3200: {1,1,1,1,1,1,1,3,3}
   5888: {1,1,1,1,1,1,1,1,9}
   8448: {1,1,1,1,1,1,1,1,2,5}
  14848: {1,1,1,1,1,1,1,1,1,10}
  23040: {1,1,1,1,1,1,1,1,1,2,2,3}
  31744: {1,1,1,1,1,1,1,1,1,1,11}
  35840: {1,1,1,1,1,1,1,1,1,1,3,4}
  39936: {1,1,1,1,1,1,1,1,1,1,2,6}
  75776: {1,1,1,1,1,1,1,1,1,1,1,12}

Length is A001222, counted by A008284, distinct A001221.
Product is A003963, counted by A339095, firsts A318871.
A similar sequence is A353503, counted by A353506.
These partitions are counted by A353698.
A005361 gives product of signature, firsts A353500 (sorted A085629).
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A353394 gives product of shadows of prime indices, firsts A353397.
Cf. A000720, A003586, A114640, A182850, A320325, A325131, A325755, A353399, A353507.

Select[Range[1000],Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]==PrimeOmega[#]&]

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A353396 Number of integer partitions of n whose Heinz number has prime shadow equal to the product of prime shadows of its parts.

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A353504 Number of integer partitions of n whose product is less than the product of their multiplicities.

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A353505 Number of integer partitions of n whose product is greater than the product of their multiplicities.

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A353699 Heinz numbers of integer partitions whose product equals their length.

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