A325244 -id:A325244 - cp's OEIS frontend

A325259 Numbers with one fewer distinct prime exponents than distinct prime factors.

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 120, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with one fewer distinct multiplicities than distinct parts. The enumeration of these partitions by sum is given by A325244.

			The sequence of terms together with their prime indices begins:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   46: {1,9}
   51: {2,7}
   55: {3,5}
   57: {2,8}
   58: {1,10}
   60: {1,1,2,3}
   62: {1,11}

Cf. A056239, A060687, A090858, A112798, A116608, A118914, A130091, A323023, A325241, A325242, A325244, A325270, A325281.

Select[Range[100],PrimeNu[#]==Length[Union[Last/@FactorInteger[#]]]+1&]

A001221(a(n)) = A071625(a(n)) + 1.

A325270 Numbers with 1 fewer distinct prime exponents than (not necessarily distinct) prime factors.

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 25, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 49, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 111, 115, 116, 117, 118, 119, 121, 122, 123, 124, 129, 133

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

Also Heinz numbers of integer partitions with 1 fewer distinct multiplicities than parts, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The enumeration of these partitions by sum is given by A117571.

Also numbers whose sorted prime signature is (1,1), (2), or (1,2). - Gus Wiseman, Jul 03 2019

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   25: {3,3}
   26: {1,6}
   28: {1,1,4}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   44: {1,1,5}

Cf. A001221, A001222, A000961, A005117, A060687, A062770, A071625, A072774, A090858, A117571, A118914, A130091, A325244, A325259.

Select[Range[100],PrimeOmega[#]==Length[Union[Last/@FactorInteger[#]]]+1&]

A378621 Antidiagonal-sums of absolute value of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

Original entry on oeis.org

1, 1, 4, 5, 11, 16, 36, 65, 142, 285, 595, 1210, 2497, 5134, 10726, 22637, 48383, 104066, 224296, 481985, 1030299, 2188912, 4626313, 9743750, 20492711, 43114180, 90843475, 191776658, 405528200, 858384333, 1817311451, 3845500427, 8129033837, 17162815092

Offset: 0

Gus Wiseman, Dec 14 2024

nonn

			Antidiagonal i + j = 3 of A175804 is (3, 1, 0, -1), so a(3) = 5.

These are the antidiagonal-sums of the absolute value of A175804.
First column of the same array is A281425.
For primes we have A376681 or A376684, signed A140119 or A376683.
For composites we have A377035, signed A377034.
For squarefree numbers we have A377040, signed A377039.
For nonsquarefree numbers we have A377048, signed A377049.
For prime powers we have A377053, signed A377052.
The signed version is A377056.
The corresponding array for strict partitions is A378622, see A293467, A377285, A378971, A378970.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A075526, A098859, A116608, A225486, A325242, A325244, A325268.

nn=30;
q=Table[PartitionsP[n],{n,0,nn}];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,Length[q]/2},{i,Length[q]/2}]
Total/@Abs/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A373241 T(n,k) is the difference between the number of different parts and the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 0, 2, 2, 1, 1, 1, 0, 0, 1, 0, 3, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Olivier Gérard, May 29 2024

This difference is always nonnegative.
The number of zero values in each row is A098859.
The number of ones in each row is A325244.
The number of positive entries in each row is A336866.
The corresponding regular triangle for partitions of n of length k is A373242.
The sum of each row is A373243.

			The array begins
  0
  0,0
  0,1,0
  0,1,0,0,0
  0,1,1,0,0,0,0
  0,1,1,0,0,2,0,0,1,0,0
  0,1,1,0,1,2,0,0,0,1,0,0,0,0,0
  0,1,1,0,1,2,0,0,2,0,1,0,0,1,1,1,0,0,0,0,0,0
  0,1,1,0,1,2,0,1,2,0,1,0,0,2,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,0
  ...

Olivier Gérard, Table of n, a(n) for n = 1..215307

Cf. A373269 a triangle of the same shape and order for number of multiplicities.

Cf. A080577, A098859, A325244, A336866, A373242, A373243.

Flatten @ Table[
  Map[Length[Union[#]] - Length[Union[Length /@ Split[#]]] &,
   IntegerPartitions[n]], {n, 1, 20}]

A325243 Number of integer partitions of n with exactly two distinct multiplicities.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 9, 12, 19, 26, 39, 47, 70, 89, 115, 148, 189, 235, 294, 362, 450, 558, 669, 817, 980, 1197, 1421, 1709, 2012, 2429, 2836, 3380, 3961, 4699, 5433, 6457, 7433, 8770, 10109, 11818, 13547, 15912, 18109, 21105, 24121, 27959, 31736, 36840, 41670

Offset: 0

Gus Wiseman, Apr 15 2019

nonn

For example, (32211) has two distinct multiplicities (1 and 2) so is counted under a(9).

The Heinz numbers of these partitions are given by A323055.

			The a(4) = 1 through a(9) = 19 partitions:
  (211)  (221)   (411)    (322)     (332)      (441)
         (311)   (3111)   (331)     (422)      (522)
         (2111)  (21111)  (511)     (611)      (711)
                          (2221)    (3221)     (3222)
                          (3211)    (4211)     (3321)
                          (4111)    (5111)     (4221)
                          (22111)   (22211)    (4311)
                          (31111)   (32111)    (5211)
                          (211111)  (41111)    (6111)
                                    (221111)   (22221)
                                    (311111)   (32211)
                                    (2111111)  (33111)
                                               (42111)
                                               (51111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)

Column k = 2 of A325242. Dominated by A325267.

Cf. A008284, A062770, A071625, A098859, A116608, A244515, A323055, A325244.

Table[Length[Select[IntegerPartitions[n],Length[Union[Length/@Split[#]]]==2&]],{n,0,30}]

A325269 Number of integer partitions of n with 2 distinct parts or at least 3 parts.

Original entry on oeis.org

0, 0, 0, 2, 3, 6, 9, 14, 20, 29, 40, 55, 75, 100, 133, 175, 229, 296, 383, 489, 625, 791, 1000, 1254, 1573, 1957, 2434, 3009, 3716, 4564, 5602, 6841, 8347, 10142, 12308, 14882, 17975, 21636, 26013, 31184, 37336, 44582, 53172, 63260, 75173, 89133, 105556

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The Heinz numbers of these partitions are given by A080257.

Partitions with 2 distinct parts are in A002133(n). Partitions with at least 3 parts are in A004250(n). Some partitions are in both subsets, so A002133(n)+A004250(n) >= a(n). - R. J. Mathar, Dec 13 2022

			The a(1) = 1 through a(8) = 20 partitions:
  (21)   (31)    (32)     (42)      (43)       (53)
  (111)  (211)   (41)     (51)      (52)       (62)
         (1111)  (221)    (222)     (61)       (71)
                 (311)    (321)     (322)      (332)
                 (2111)   (411)     (331)      (422)
                 (11111)  (2211)    (421)      (431)
                          (3111)    (511)      (521)
                          (21111)   (2221)     (611)
                          (111111)  (3211)     (2222)
                                    (4111)     (3221)
                                    (22111)    (3311)
                                    (31111)    (4211)
                                    (211111)   (5111)
                                    (1111111)  (22211)
                                               (32111)
                                               (41111)
                                               (221111)
                                               (311111)
                                               (2111111)
                                               (11111111)

Cf. A001221, A001222, A001358, A001399, A007774, A008284, A060687, A080257, A090858, A116608, A325244.

Cf. A000041, A002133, A004250.

A325269 := proc(n)
    local a,p,s ;
    a := 0 ;
    for p in combinat[partition](n) do
        s := convert(p,set) ;
        if nops(p) >= 3 or nops(s) = 2 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A325269(n),n=0..40) ; # R. J. Mathar, Dec 13 2022

Table[Length[Select[IntegerPartitions[n],Length[Union[#]]==2||Length[#]>2&]],{n,0,30}]

conjecture: a(n) = A000041(n) - A000034(n-1), n>0. - R. J. Mathar, Dec 13 2022

A325279 Number of integer partitions of n whose maximum multiplicity is one greater than their minimum multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 5, 6, 9, 10, 18, 18, 31, 34, 48, 57, 80, 86, 122, 138, 183, 211, 275, 311, 402, 461, 576, 663, 825, 942, 1163, 1334, 1621, 1865, 2248, 2566, 3084, 3532, 4193, 4794, 5674, 6472, 7617, 8685, 10153, 11576, 13483, 15320, 17790, 20200, 23342

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The Heinz numbers of these partitions are given by A325241.

For example, the partition (44111) has two multiplicities (2 and 3) which differ by 1, so is counted under a(11).

			The a(4) = 1 through a(11) = 18 partitions:
  (211)  (221)  (411)  (322)    (332)    (441)    (433)      (443)
         (311)         (331)    (422)    (522)    (442)      (533)
                       (511)    (611)    (711)    (622)      (551)
                       (3211)   (3221)   (3321)   (811)      (722)
                       (22111)  (4211)   (4221)   (5221)     (911)
                                (22211)  (4311)   (5311)     (4322)
                                         (5211)   (6211)     (4331)
                                         (32211)  (33211)    (4421)
                                         (33111)  (42211)    (5411)
                                                  (2221111)  (6221)
                                                             (6311)
                                                             (7211)
                                                             (33221)
                                                             (33311)
                                                             (43211)
                                                             (44111)
                                                             (52211)
                                                             (2222111)

Cf. A047966, A062770, A071625, A098859, A117571, A127002, A325242, A325244, A325245, A325280.

Table[Length[Select[IntegerPartitions[n],Max@@Length/@Split[#]-Min@@Length/@Split[#]==1&]],{n,0,30}]

cp's OEIS Frontend

A325259 Numbers with one fewer distinct prime exponents than distinct prime factors.

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A325270 Numbers with 1 fewer distinct prime exponents than (not necessarily distinct) prime factors.

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A378621 Antidiagonal-sums of absolute value of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

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A373241 T(n,k) is the difference between the number of different parts and the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).

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A325243 Number of integer partitions of n with exactly two distinct multiplicities.

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A325269 Number of integer partitions of n with 2 distinct parts or at least 3 parts.

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Maple

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A325279 Number of integer partitions of n whose maximum multiplicity is one greater than their minimum multiplicity.

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