A325041 -id:A325041 - cp's OEIS frontend

A325037 Heinz numbers of integer partitions whose product of parts is greater than their sum.

1, 15, 21, 25, 27, 33, 35, 39, 42, 45, 49, 50, 51, 54, 55, 57, 63, 65, 66, 69, 70, 75, 77, 78, 81, 85, 87, 90, 91, 93, 95, 98, 99, 100, 102, 105, 110, 111, 114, 115, 117, 119, 121, 123, 125, 126, 129, 130, 132, 133, 135, 138, 140, 141, 143, 145, 147, 150, 153

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is greater than their sum of prime indices (A056239).

The enumeration of these partitions by sum is given by A114324.

			The sequence of terms together with their prime indices begins:
   1: {}
  15: {2,3}
  21: {2,4}
  25: {3,3}
  27: {2,2,2}
  33: {2,5}
  35: {3,4}
  39: {2,6}
  42: {1,2,4}
  45: {2,2,3}
  49: {4,4}
  50: {1,3,3}
  51: {2,7}
  54: {1,2,2,2}
  55: {3,5}
  57: {2,8}
  63: {2,2,4}
  65: {3,6}
  66: {1,2,5}
  69: {2,9}
  70: {1,3,4}
  75: {2,3,3}
  77: {4,5}
  78: {1,2,6}
  81: {2,2,2,2}

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325038, A325041, A325042, A325044.

q:= n-> (l-> mul(i, i=l)>add(i, i=l))(map(i->
    numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..200])[];  # Alois P. Heinz, Mar 27 2019

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Times@@primeMS[#]>Plus@@primeMS[#]&]

A003963(a(n)) > A056239(a(n)).

A325044 Heinz numbers of integer partitions whose sum of parts is greater than or equal to their product.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 34, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 52, 53, 56, 58, 59, 60, 61, 62, 64, 67, 68, 71, 72, 73, 74, 76, 79, 80, 82, 83, 84, 86, 88, 89, 92, 94, 96, 97, 101

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is less than or equal to their sum of prime indices (A056239).

The enumeration of these partitions by sum is given by A096276.

			The sequence of terms together with their prime indices begins:
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   6: {1,2}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  22: {1,5}
  23: {9}
  24: {1,1,1,2}

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325038, A325041, A325042.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Times@@primeMS[#]<=Plus@@primeMS[#]&]

A003963(a(n)) <= A056239(a(n)).
a(n) = A325038(n)/2.
Union of A301987 and A325038.

A325038 Heinz numbers of integer partitions whose sum of parts is greater than their product.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 36, 38, 40, 44, 46, 48, 52, 56, 58, 60, 62, 64, 68, 72, 74, 76, 80, 82, 86, 88, 92, 94, 96, 104, 106, 112, 116, 118, 120, 122, 124, 128, 134, 136, 142, 144, 146, 148, 152, 158, 160, 164, 166, 168, 172

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is less than their sum of prime indices (A056239).

The enumeration of these partitions by sum is given by A096276 shifted once to the right.

			The sequence of terms together with their prime indices begins:
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  32: {1,1,1,1,1}
  34: {1,7}
  36: {1,1,2,2}
  38: {1,8}
  40: {1,1,1,3}
  44: {1,1,5}
  46: {1,9}
  48: {1,1,1,1,2}

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325041, A325042, A325044.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Times@@primeMS[#]

A003963(a(n)) < A056239(a(n)).

a(n) = 2 * A325044(n).

A325036 Difference between product and sum of prime indices of n.

Original entry on oeis.org

1, 0, 0, -1, 0, -1, 0, -2, 0, -1, 0, -2, 0, -1, 1, -3, 0, -1, 0, -2, 2, -1, 0, -3, 3, -1, 2, -2, 0, 0, 0, -4, 3, -1, 5, -2, 0, -1, 4, -3, 0, 1, 0, -2, 5, -1, 0, -4, 8, 2, 5, -2, 0, 1, 7, -3, 6, -1, 0, -1, 0, -1, 8, -5, 9, 2, 0, -2, 7, 4, 0, -3, 0, -1, 10, -2, 11, 3, 0, -4, 8, -1, 0, 0, 11, -1, 8, -3, 0, 4, 14, -2, 9

Offset: 1

Gus Wiseman, Mar 25 2019

sign

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.

			The prime indices of 45 are {2,2,3}, with product 12 and sum 7, so a(45) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Positions of zeros are A301987. Positions of ones are A325041. Positions of negative ones are A325042.
Cf. A000720, A001222, A003963, A056239, A112798, A122111, A178503, A175508, A319000.
Cf. A325032, A325033, A325034, A325035, A325037, A325038, A325040, A325044.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@primeMS[n]-Total[primeMS[n]],{n,100}]
dps[n_]:=Module[{pi=Flatten[Table[PrimePi[#[[1]]],#[[2]]]&/@FactorInteger[n]]},Times@@pi-Total[pi]]; Join[{1},Array[dps,100,2]] (* Harvey P. Dale, May 26 2023 *)

A003963(n) = { n=factor(n); n[, 1]=apply(primepi, n[, 1]); factorback(n) }; \\ From A003963
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); };
A325036(n) = (A003963(n) - A056239(n)); \\ Antti Karttunen, May 08 2022

a(n) = A003963(n) - A056239(n).

For all n >= 1, a(A325040(n)) = a(A122111(A325040(n))). - Antti Karttunen, May 08 2022

Data section extended up to a(93) by Antti Karttunen, May 08 2022

A379722 Numbers whose prime indices do not have the same sum as product.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 85, 86, 87, 88, 90, 91, 92, 93

Offset: 1

Gus Wiseman, Jan 08 2025

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.
Partitions of this type are counted by A379736.
The complement is A301987, counted by A001055.

			The terms together with their prime indices begin:
    1: {}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}

Nonzeros of A325036.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A324851 finds numbers > 1 divisible by the sum of their prime indices.
A379666 counts partitions by sum and product, without 1's A379668.
A379681 gives sum plus product of prime indices, firsts A379682.
Counting and ranking multisets by comparing sum and product:
- same: A001055 (strict A045778), ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721
- different: A379736, ranks A379722 (this)
Cf. A000720, A003963, A025147, A075254, A111133, A178503, A319000, A325041.

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

A379736 Number of integer partitions of n whose product of parts is not n.

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 9, 14, 19, 28, 40, 55, 73, 100, 133, 174, 226, 296, 381, 489, 623, 790, 1000, 1254, 1568, 1956, 2434, 3007, 3714, 4564, 5599, 6841, 8342, 10141, 12308, 14881, 17968, 21636, 26013, 31183, 37331, 44582, 53169, 63260, 75171, 89130, 105556

Offset: 0

Gus Wiseman, Jan 07 2025

nonn

These partitions are ranked by A379722, complement A301987.

			The a(2) = 1 through a(7) = 14 partitions:
  (11)  (21)   (31)    (32)     (33)      (43)
        (111)  (211)   (41)     (42)      (52)
               (1111)  (221)    (51)      (61)
                       (311)    (222)     (322)
                       (2111)   (411)     (331)
                       (11111)  (2211)    (421)
                                (3111)    (511)
                                (21111)   (2221)
                                (111111)  (3211)
                                          (4111)
                                          (22111)
                                          (31111)
                                          (211111)
                                          (1111111)

The complement is counted by A001055.
The strict case is A111133 (except first term).
A000041 counts integer partitions, strict A000009.
A002865 counts partitions into parts > 1, see A379734, strict A379735.
A324851 finds numbers > 1 divisible by the sum of their prime indices.
A379666 counts partitions by sum and product, without 1's A379668.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736 (this), ranks A379722
Cf. A028422, A069016, A319000, A319916, A325036, A325041, A326152, A379671, A379678.

Table[Length[Select[IntegerPartitions[n],Times@@#!=n&]],{n,0,30}]

a(n) = A000041(n) - A001055(n).

A379666 Array read by antidiagonals downward where A(n,k) is the number of integer partitions of n with product k.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 01 2025

Counts finite multisets of positive integers by sum and product.

			Array begins:
        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
        -----------------------------------------------
   n=0:  1   0   0   0   0   0   0   0   0   0   0   0
   n=1:  1   0   0   0   0   0   0   0   0   0   0   0
   n=2:  1   1   0   0   0   0   0   0   0   0   0   0
   n=3:  1   1   1   0   0   0   0   0   0   0   0   0
   n=4:  1   1   1   2   0   0   0   0   0   0   0   0
   n=5:  1   1   1   2   1   1   0   0   0   0   0   0
   n=6:  1   1   1   2   1   2   0   2   1   0   0   0
   n=7:  1   1   1   2   1   2   1   2   1   1   0   2
   n=8:  1   1   1   2   1   2   1   3   1   1   0   3
   n=9:  1   1   1   2   1   2   1   3   2   1   0   3
  n=10:  1   1   1   2   1   2   1   3   2   2   0   3
  n=11:  1   1   1   2   1   2   1   3   2   2   1   3
  n=12:  1   1   1   2   1   2   1   3   2   2   1   4
For example, the A(9,12) = 3 partitions are: (6,2,1), (4,3,1,1), (3,2,2,1,1).
Antidiagonals begin:
   n+k=1: 1
   n+k=2: 0 1
   n+k=3: 0 0 1
   n+k=4: 0 0 1 1
   n+k=5: 0 0 0 1 1
   n+k=6: 0 0 0 1 1 1
   n+k=7: 0 0 0 0 1 1 1
   n+k=8: 0 0 0 0 2 1 1 1
   n+k=9: 0 0 0 0 0 2 1 1 1
  n+k=10: 0 0 0 0 0 1 2 1 1 1
  n+k=11: 0 0 0 0 0 1 1 2 1 1 1
  n+k=12: 0 0 0 0 0 0 2 1 2 1 1 1
  n+k=13: 0 0 0 0 0 0 0 2 1 2 1 1 1
  n+k=14: 0 0 0 0 0 0 2 1 2 1 2 1 1 1
  n+k=15: 0 0 0 0 0 0 1 2 1 2 1 2 1 1 1
  n+k=16: 0 0 0 0 0 0 0 1 3 1 2 1 2 1 1 1
For example, antidiagonal n+k=10 counts the following partitions:
  n=5: (5)
  n=6: (411), (2211)
  n=7: (31111)
  n=8: (2111111)
  n=9: (111111111)
so the 10th antidiagonal is: (0,0,0,0,0,1,2,1,1,1).

Row sums are A000041 = partitions of n, strict A000009, no ones A002865.
Diagonal A(n,n) is A001055(n) = factorizations of n, strict A045778.
Antidiagonal sums are A379667.
The case without ones is A379668, antidiagonal sums A379669 (zeros A379670).
The strict case is A379671, antidiagonal sums A379672.
The strict case without ones is A379678, antidiagonal sums A379679 (zeros A379680).
A316439 counts factorizations by length, partitions A008284.
A326622 counts factorizations with integer mean, strict A328966.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A003963, A028422, A069016, A096765, A318950, A319000, A319916, A319057, A325036, A325041, A325042, A326152.

nn=12;
tt=Table[Length[Select[IntegerPartitions[n],Times@@#==k&]],{n,0,nn},{k,1,nn}] (* array *)
tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
Join@@tr (* sequence *)

A379720 Except a(0)=1 and a(4)=0, number of integer partitions of n with no 1's and at least two parts.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 3, 3, 6, 7, 11, 13, 20, 23, 33, 40, 54, 65, 87, 104, 136, 164, 209, 252, 319, 382, 477, 573, 707, 846, 1038, 1237, 1506, 1793, 2166, 2572, 3093, 3659, 4377, 5169, 6152, 7244, 8590, 10086, 11913, 13958, 16423, 19195, 22518, 26251, 30700, 35716

Offset: 0

Gus Wiseman, Jan 06 2025

nonn

Also partitions of n such that all parts are > 1 and have product > n.

Allowing 1's gives A114324, ranks A325037. The strict case is A318029 (except first term).

			The a(5) = 1 through a(11) = 13 partitions:
  (3,2)  (3,3)    (4,3)    (4,4)      (5,4)      (5,5)        (6,5)
         (4,2)    (5,2)    (5,3)      (6,3)      (6,4)        (7,4)
         (2,2,2)  (3,2,2)  (6,2)      (7,2)      (7,3)        (8,3)
                           (3,3,2)    (3,3,3)    (8,2)        (9,2)
                           (4,2,2)    (4,3,2)    (4,3,3)      (4,4,3)
                           (2,2,2,2)  (5,2,2)    (4,4,2)      (5,3,3)
                                      (3,2,2,2)  (5,3,2)      (5,4,2)
                                                 (6,2,2)      (6,3,2)
                                                 (3,3,2,2)    (7,2,2)
                                                 (4,2,2,2)    (3,3,3,2)
                                                 (2,2,2,2,2)  (4,3,2,2)
                                                              (5,2,2,2)
                                                              (3,2,2,2,2)

For <= instead of < we have A002865 = partitions into parts > 1.
These partitions have ranks A071904 (except initial terms).
Set a(4) = 1 to get A083751.
A000041 counts integer partitions, strict A000009.
A379668 counts partitions without 1's by sum and product.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A003963, A028422, A318950, A319000, A319916, A325036, A325041, A326152, A326178, A379666, A379678.

Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&Plus@@#

Except for n = 0 and n = 4, we have a(n) = A002865(n) - 1.

A379671 Array read by antidiagonals downward where A(n,k) is the number of finite sets of positive integers with sum n and product k.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 01 2025

Counts finite sets of positive integers by sum and product.

			Array begins:
        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
        -----------------------------------------------
   n=0:  1   0   0   0   0   0   0   0   0   0   0   0
   n=1:  1   0   0   0   0   0   0   0   0   0   0   0
   n=2:  0   1   0   0   0   0   0   0   0   0   0   0
   n=3:  0   1   1   0   0   0   0   0   0   0   0   0
   n=4:  0   0   1   1   0   0   0   0   0   0   0   0
   n=5:  0   0   0   1   1   1   0   0   0   0   0   0
   n=6:  0   0   0   0   1   2   0   1   0   0   0   0
   n=7:  0   0   0   0   0   1   1   1   0   1   0   1
   n=8:  0   0   0   0   0   0   1   1   0   1   0   2
   n=9:  0   0   0   0   0   0   0   1   1   0   0   1
  n=10:  0   0   0   0   0   0   0   0   1   1   0   0
  n=11:  0   0   0   0   0   0   0   0   0   1   1   0
  n=12:  0   0   0   0   0   0   0   0   0   0   1   1
The A(8,12) = 2 sets are: {2,6}, {1,3,4}.
The A(14,40) = 2 sets are: {4,10}, {1,5,8}.
Antidiagonals begin:
   n+k=1: 1
   n+k=2: 0 1
   n+k=3: 0 0 0
   n+k=4: 0 0 1 0
   n+k=5: 0 0 0 1 0
   n+k=6: 0 0 0 1 0 0
   n+k=7: 0 0 0 0 1 0 0
   n+k=8: 0 0 0 0 1 0 0 0
   n+k=9: 0 0 0 0 0 1 0 0 0
  n+k=10: 0 0 0 0 0 1 0 0 0 0
  n+k=11: 0 0 0 0 0 1 1 0 0 0 0
  n+k=12: 0 0 0 0 0 0 2 0 0 0 0 0
  n+k=13: 0 0 0 0 0 0 0 1 0 0 0 0 0
  n+k=14: 0 0 0 0 0 0 1 1 0 0 0 0 0 0
  n+k=15: 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0
  n+k=16: 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
For example, antidiagonal n+k=11 counts the following sets:
  n=5: {2,3}
  n=6: {1,5}
so the 11th antidiagonal is: (0,0,0,0,0,1,1,0,0,0,0).

Row sums are A000009 = strict partitions, non-strict A000041.
Column sums are 2*A045778 where A045778 = strict factorizations, non-strict A001055.
Antidiagonal sums are A379672, non-strict A379667 (zeros A379670).
Without ones we have A379678, antidiagonal sums A379679 (zeros A379680).
The non-strict version is A379666, without ones A379668.
A316439 counts factorizations by length, partitions A008284.
A326622 counts factorizations with integer mean, strict A328966.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A002865, A003963, A028422, A069016, A318950, A319000, A319916, A319057, A325036, A325041, A325042, A326152.

nn=12;
tt=Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Times@@#==k&]],{n,0,nn},{k,1,nn}] (* array *)
tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
Join@@tr (* sequence *)

A325042 Heinz numbers of integer partitions whose product of parts is one fewer than their sum.

Original entry on oeis.org

4, 6, 10, 14, 18, 22, 26, 34, 38, 46, 58, 60, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 168, 178, 194, 202, 206, 214, 216, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 400, 422, 446, 454, 458, 466

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is one fewer than their sum of prime indices (A056239).

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    6: {1,2}
   10: {1,3}
   14: {1,4}
   18: {1,2,2}
   22: {1,5}
   26: {1,6}
   34: {1,7}
   38: {1,8}
   46: {1,9}
   58: {1,10}
   60: {1,1,2,3}
   62: {1,11}
   74: {1,12}
   82: {1,13}
   86: {1,14}
   94: {1,15}
  106: {1,16}
  118: {1,17}
  122: {1,18}

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325038, A325041, A325044.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Times@@primeMS[#]==Total[primeMS[#]]-1&]

A003963(a(n)) = A056239(a(n)) - 1.

a(n) = 2 * A301987(n).

cp's OEIS Frontend

A325037 Heinz numbers of integer partitions whose product of parts is greater than their sum.

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A325044 Heinz numbers of integer partitions whose sum of parts is greater than or equal to their product.

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A325038 Heinz numbers of integer partitions whose sum of parts is greater than their product.

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A325036 Difference between product and sum of prime indices of n.

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A379722 Numbers whose prime indices do not have the same sum as product.

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A379736 Number of integer partitions of n whose product of parts is not n.

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A379666 Array read by antidiagonals downward where A(n,k) is the number of integer partitions of n with product k.

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A379720 Except a(0)=1 and a(4)=0, number of integer partitions of n with no 1's and at least two parts.

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