A324851 -id:A324851 - cp's OEIS frontend

A331384 Numbers whose sum of prime factors is equal to their product of prime indices.

35, 65, 95, 98, 154, 324, 364, 476, 623, 763, 791, 812, 826, 938, 994, 1036, 1064, 1106, 1144, 1148, 1162, 1288, 1484, 1708, 1736, 2044, 2408, 2632, 4320, 5408, 6688, 6974, 8000, 10208, 12623, 12701, 12779, 14144, 19624, 23144, 25784, 26048, 44176, 47696

Offset: 1

Gus Wiseman, Jan 16 2020

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.

Numbers k such that A001414(k) = A003963(k). - Jason Yuen, Dec 19 2024

			The sequence of terms together with their prime indices begins:
     35: {3,4}
     65: {3,6}
     95: {3,8}
     98: {1,4,4}
    154: {1,4,5}
    324: {1,1,2,2,2,2}
    364: {1,1,4,6}
    476: {1,1,4,7}
    623: {4,24}
    763: {4,29}
    791: {4,30}
    812: {1,1,4,10}
    826: {1,4,17}
    938: {1,4,19}
    994: {1,4,20}
   1036: {1,1,4,12}
   1064: {1,1,1,4,8}
   1106: {1,4,22}
   1144: {1,1,1,5,6}
   1148: {1,1,4,13}
For example, 476 has prime factors {2,2,7,17} and prime indices {1,1,4,7}, and 2+2+7+17 = 28 = 1*1*4*7, so 476 is in the sequence.

These are the Heinz numbers of the partitions counted by A331383.
Numbers divisible by the sum of their prime factors are A036844.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Product of prime indices is divisible by sum of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Partitions whose Heinz number is divisible by their sum of primes: A330953.
Sum of prime factors is divisible by sum of prime indices: A331380
Partitions whose product divides their sum of primes are A331381.
Cf. A000040, A001414, A003963, A324850, A330954, A331378, A331379, A331382, A331415, A331416.

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

A379734 Number of integer partitions of n into parts > 1 whose product is a multiple of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 4, 3, 2, 1, 8, 1, 4, 8, 27, 1, 32, 1, 40, 24, 13, 1, 175, 56, 22, 188, 166, 1, 387, 1, 874, 166, 61, 410, 1833, 1, 98, 391, 3028, 1, 2704, 1, 1828, 5893, 239, 1, 16756, 3446, 9742, 1865, 5276, 1, 32927, 8179, 31643, 3840, 814, 1, 82958, 1

Offset: 1

Gus Wiseman, Jan 07 2025

nonn

Allowing 1's gives A057568.

			The a(n) partitions for n = 6, 10, 12, 15, 22:
  (6)  (10)     (12)         (15)         (22)
       (5,3,2)  (6,6)        (6,5,4)      (11,6,5)
                (5,4,3)      (7,5,3)      (11,7,4)
                (6,4,2)      (10,3,2)     (11,8,3)
                (4,3,3,2)    (5,4,3,3)    (11,9,2)
                (5,3,2,2)    (5,5,3,2)    (11,4,4,3)
                (6,2,2,2)    (6,5,2,2)    (11,5,4,2)
                (3,3,2,2,2)  (5,3,3,2,2)  (11,6,3,2)
                                          (11,7,2,2)
                                          (11,3,3,3,2)
                                          (11,4,3,2,2)
                                          (11,5,2,2,2)
                                          (11,3,2,2,2,2)

These partitions are ranked by the odd terms of A326149.
The strict version is A379735, allowing 1's A379733.
A000041 counts integer partitions, strict A000009.
A002865 counts partitions into parts > 1.
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, ranks A379722, see A111133
Cf. A069016, A318950, A319000, A319916, A324851, A325041, A326152, A379671, A379678.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=1, 1, 0), `if`(i<2, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, min(i, n-i), t/igcd(i, t)))))
    end:
a:= n-> `if`(isprime(n), 1, b(n$3)):
seq(a(n), n=1..70);  # Alois P. Heinz, Jan 07 2025

Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&Divisible[Times@@#,n]&]],{n,30}]

A340828 Number of strict integer partitions of n whose maximum part is a multiple of their length.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 2, 4, 5, 6, 6, 7, 8, 11, 10, 13, 17, 18, 21, 24, 27, 30, 35, 39, 46, 53, 61, 68, 79, 87, 97, 110, 123, 139, 157, 175, 196, 222, 247, 278, 312, 347, 385, 433, 476, 531, 586, 651, 720, 800, 883, 979, 1085, 1200, 1325, 1464, 1614, 1777

Offset: 1

Gus Wiseman, Feb 01 2021

nonn

			The a(1) = 1 through a(16) = 10 partitions (A..G = 10..16):
  1  2  3   4  5   6    7   8   9    A     B    C    D    E     F      G
        21     41  42   43  62  63   64    65   84   85   86    87     A6
                   321  61      81   82    83   A2   A3   A4    A5     C4
                                621  631   A1   642  C1   C2    C3     E2
                                     4321  632  651  643  653   E1     943
                                           641  921  652  932   654    952
                                                     931  941   942    961
                                                          8321  951    C31
                                                                C21    8431
                                                                8421   8521
                                                                54321

FindStat, St000010: The length of the partition.
FindStat, St000147: The largest part of an integer partition.
FindStat, St000784: The maximum of the length and the largest part of the integer partition.

Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict version is A168659 (A340609/A340610).
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A064173 counts partitions of positive/negative rank (A340787/A340788).
A067538 counts partitions whose length/max divides sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A096401 counts strict partition with length equal to minimum.
A102627 counts strict partitions with length dividing sum.
A326842 counts partitions whose length and parts all divide sum (A326847).
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A340829 counts strict partitions with Heinz number divisible by sum.
A340830 counts strict partitions with all parts divisible by length.
Cf. A003114, A006141, A064174, A117409, A143773 (A316428), A200750, A326843 (A326837), A330950 (A324851).

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Max@@#,Length[#]]&]],{n,30}]

A340830 Number of strict integer partitions of n such that every part is a multiple of the number of parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 4, 1, 4, 1, 6, 1, 5, 2, 6, 1, 8, 1, 7, 4, 7, 1, 12, 1, 8, 6, 9, 1, 16, 1, 10, 9, 11, 1, 21, 1, 12, 13, 12, 1, 28, 1, 13, 17, 16, 1, 33, 1, 19, 22, 15, 1, 45, 1, 16, 28, 25, 1, 47, 1, 28, 34, 18

Offset: 1

Gus Wiseman, Feb 02 2021

nonn

			The a(n) partitions for n = 1, 6, 10, 14, 18, 20, 24, 26, 30:
  1   6     10    14     18      20     24       26      30
      4,2   6,4   8,6    10,8    12,8   16,8     18,8    22,8
            8,2   10,4   12,6    14,6   18,6     20,6    24,6
                  12,2   14,4    16,4   20,4     22,4    26,4
                         16,2    18,2   22,2     24,2    28,2
                         9,6,3          14,10    14,12   16,14
                                        12,9,3   16,10   18,12
                                        15,6,3           20,10
                                                         15,9,6
                                                         18,9,3
                                                         21,6,3
                                                         15,12,3

Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict case is A143773 (A316428).
The case where length divides sum also is A340827.
The version for factorizations is A340851.
Factorization of this type are counted by A340853.
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions whose length/max divide sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A102627 counts strict partitions whose length divides sum.
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A340828 counts strict partitions with length divisible by maximum.
A340829 counts strict partitions with Heinz number divisible by sum.
Cf. A114638, A168659, A326641, A326843 (A326837), A326849, A326852 (A326838), A330950 (A324851), A340852.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@IntegerQ/@(#/Length[#])&]],{n,30}]

a(n) = Sum_{d|n} A008289(n/d, d).

A379319 Even numbers whose product of prime indices is a multiple of their sum of prime indices.

Original entry on oeis.org

2, 30, 84, 108, 150, 154, 190, 198, 200, 264, 364, 390, 442, 468, 490, 506, 580, 624, 630, 658, 700, 714, 810, 840, 846, 874, 900, 918, 952, 988, 1020, 1080, 1110, 1118, 1120, 1224, 1254, 1330, 1430, 1440, 1480, 1596, 1632, 1666, 1708, 1710, 1716, 1786, 1794

Offset: 1

Gus Wiseman, Jan 17 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. The sum and product of prime indices are A056239 and A003963 respectively.

			The prime indices of 150 are {1,2,3,3}, with sum 9 and product 18, so 150 is in the sequence.
The terms together with their prime indices begin:
     2: {1}
    30: {1,2,3}
    84: {1,1,2,4}
   108: {1,1,2,2,2}
   150: {1,2,3,3}
   154: {1,4,5}
   190: {1,3,8}
   198: {1,2,2,5}
   200: {1,1,1,3,3}
   264: {1,1,1,2,5}
   364: {1,1,4,6}
   390: {1,2,3,6}
   442: {1,6,7}
   468: {1,1,2,2,6}
   490: {1,3,4,4}

Even terms of A326149, which is counted by A057568 (strict A379733).
For nonprime instead of even we have A326150.
For odd instead of even we have A379318, counted by A379734 (strict A379735).
Partitions of this type are counted by A379320.
For squarefree instead of even we have A379844.
The squarefree case is A379845.
Divide all terms by 2 to get A380217.
A000040 lists the prime numbers, differences A001223.
A003963 multiplies together prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- 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, ranks A379722, see A111133
Cf. A069016, A301988, A318950, A319000, A324851, A325041, A379671, A379678.

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

A379735 Number of strict integer partitions of n into parts > 1 whose product is a multiple of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 3, 4, 4, 1, 8, 1, 11, 9, 7, 1, 26, 7, 10, 18, 33, 1, 67, 1, 56, 37, 20, 69, 158, 1, 27, 70, 252, 1, 280, 1, 207, 402, 52, 1, 834, 133, 423, 226, 465, 1, 1132, 635, 1541, 388, 129, 1, 3377, 1, 171, 2891, 3561, 1674, 3154

Offset: 1

Gus Wiseman, Jan 07 2025

nonn

These partitions are ranked by the odd squarefree terms of A326149.

			The a(n) partitions for n = 2, 9, 12, 15, 18, 20, 21:
  (2)  (9)    (12)     (15)      (18)       (20)         (21)
       (6,3)  (5,4,3)  (6,5,4)   (12,6)     (8,7,5)      (8,7,6)
              (6,4,2)  (7,5,3)   (9,5,4)    (10,6,4)     (9,7,5)
                       (10,3,2)  (9,6,3)    (10,8,2)     (11,7,3)
                                 (9,7,2)    (11,5,4)     (12,7,2)
                                 (6,5,4,3)  (12,5,3)     (14,4,3)
                                 (7,6,3,2)  (7,6,5,2)    (7,6,5,3)
                                 (9,4,3,2)  (8,5,4,3)    (9,7,3,2)
                                            (9,5,4,2)    (7,5,4,3,2)
                                            (10,5,3,2)
                                            (6,5,4,3,2)

Allowing 1's gives A379733.
The non-strict version is A379734, allowing 1's A057568.
A000041 counts integer partitions, strict A000009.
A002865 counts partitions into parts > 1.
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, ranks A379722, see A111133
Cf. A069016, A319000, A319057, A319916, A324851, A326152, A379671, A379678.

Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&UnsameQ@@#&&Divisible[Times@@#,n]&]],{n,30}]

A331380 Numbers whose sum of prime factors is divisible by their sum of prime indices.

Original entry on oeis.org

2, 4, 8, 16, 32, 33, 39, 55, 64, 65, 66, 74, 77, 78, 86, 91, 110, 128, 130, 132, 154, 156, 164, 182, 188, 220, 256, 260, 264, 308, 312, 364, 371, 411, 440, 459, 512, 513, 520, 528, 530, 616, 624, 636, 689, 728, 746, 755, 765, 766, 855, 880, 906, 915, 918, 1007

Offset: 1

Gus Wiseman, Jan 16 2020

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.

			The sequence of terms together with their prime indices begins:
     2: {1}
     4: {1,1}
     8: {1,1,1}
    16: {1,1,1,1}
    32: {1,1,1,1,1}
    33: {2,5}
    39: {2,6}
    55: {3,5}
    64: {1,1,1,1,1,1}
    65: {3,6}
    66: {1,2,5}
    74: {1,12}
    77: {4,5}
    78: {1,2,6}
    86: {1,14}
    91: {4,6}
   110: {1,3,5}
   128: {1,1,1,1,1,1,1}
   130: {1,3,6}
   132: {1,1,2,5}
For example, 132 has prime factors {2,2,3,11} and prime indices {1,1,2,5}, and 18 is divisible by 9, so 132 is in the sequence.

These are the Heinz numbers of the partitions counted by A331380.
Numbers divisible by the sum of their prime factors are A036844.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Product of prime indices is divisible by sum of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Heinz number is divisible by sum of primes: A330953.
Partitions whose product divides their sum of primes are A331381.
Partitions whose product is equal to their sum of primes are A331383.
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A001414, A056239, A330954, A331378, A331379, A331382, A331415, A331416.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],Divisible[Plus@@Prime/@primeMS[#],Plus@@primeMS[#]]&]

A340827 Number of strict integer partitions of n into divisors of n whose length also divides n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 18, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 17, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1, 12, 1, 1, 1, 3, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 01 2021

nonn

The first element not in A326715 that is however a Heinz number of these partitions is 273.

			The a(n) partitions for n = 6, 12, 24, 90, 84:
  6       12        24            90                      84
  3,2,1   6,4,2     12,8,4        45,30,15                42,28,14
          6,3,2,1   12,6,4,2      45,30,9,5,1             42,21,14,7
                    12,8,3,1      45,18,15,9,3            42,28,12,2
                    8,6,4,3,2,1   45,30,10,3,2            42,28,6,4,3,1
                                  45,18,15,10,2           42,28,7,4,2,1
                                  45,30,6,5,3,1           42,14,12,7,6,3
                                  45,30,9,3,2,1           42,21,12,4,3,2
                                  45,15,10,9,6,5          42,21,12,6,2,1
                                  45,18,10,9,5,3          42,21,14,4,2,1
                                  45,18,10,9,6,2          28,21,14,12,6,3
                                  45,18,15,6,5,1          28,21,14,12,7,2
                                  45,18,15,9,2,1          42,21,7,6,4,3,1
                                  30,18,15,10,6,5,3,2,1   42,14,12,7,4,3,2
                                                          42,14,12,7,6,2,1
                                                          28,21,14,12,4,3,2
                                                          28,21,14,12,6,2,1

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 2519 terms from Antti Karttunen)
Antti Karttunen, Scheme program for computing this sequence

Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict case is A326842 (A326847).
A018818 = partitions using divisors (A326841).
A047993 = balanced partitions (A106529).
A067538 = partitions whose length/maximum divides sum (A316413/A326836).
A072233 = partitions by sum and length, with strict case A008289.
A102627 = strict partitions whose length divides sum.
A326850 = strict partitions whose maximum part divides sum.
A326851 = strict partitions w/ length and max dividing sum.
A340828 = strict partitions w/ length divisible by max.
A340829 = strict partitions w/ Heinz number divisible by sum.
A340830 = strict partitions w/ parts divisible by length.
Cf. A064173, A143773 (A316428), A168659, A326843 (A326837), A326852, A330950 (A324851), A340851, A340853.

Table[Length[Select[IntegerPartitions[n,All,Divisors[n]],UnsameQ@@#&&Divisible[n,Length[#]]&]],{n,30}]

A340827(n, divsleft=List(divisors(n)), rest=n, len=0) = if(rest<=0, !rest && !(n%len), my(s=0, d); forstep(i=#divsleft, 1, -1, d = divsleft[i]; listpop(divsleft,i); if(rest>=d, s += A340827(n, divsleft, rest-d, 1+len))); (s)); \\ Antti Karttunen, Feb 22 2023

;; See the Links-section. - Antti Karttunen, Feb 22 2023

Data section extended up to a(105) by Antti Karttunen, Feb 22 2023

A331382 Numbers whose sum of prime factors is divisible by their product of prime indices.

Original entry on oeis.org

1, 2, 4, 8, 16, 18, 20, 32, 35, 44, 60, 62, 64, 65, 68, 72, 92, 95, 98, 128, 154, 160, 168, 256, 264, 288, 291, 303, 324, 364, 400, 476, 480, 512, 618, 623, 624, 642, 706, 763, 791, 812, 816, 826, 938, 994, 1024, 1036, 1064, 1068, 1106, 1144, 1148, 1152, 1162

Offset: 1

Gus Wiseman, Jan 16 2020

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.

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     8: {1,1,1}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    32: {1,1,1,1,1}
    35: {3,4}
    44: {1,1,5}
    60: {1,1,2,3}
    62: {1,11}
    64: {1,1,1,1,1,1}
    65: {3,6}
    68: {1,1,7}
    72: {1,1,1,2,2}
    92: {1,1,9}
    95: {3,8}
    98: {1,4,4}
   128: {1,1,1,1,1,1,1}
For example, 60 has prime factors {2,2,3,5} and prime indices {1,1,2,3}, and 12 is divisible by 6, so 60 is in the sequence.

These are the Heinz numbers of the partitions counted by A331381.
Numbers divisible by the sum of their prime factors are A036844.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Product of prime indices is divisible by sum of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Sum of prime factors is divisible by sum of prime indices: A331380
Partitions whose product is equal to the sum of primes are A331383.
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A001414, A324850, A330953, A330954, A331378, A331379.

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

A379320 Number of integer partitions of n whose product is a multiple of n + 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 2, 3, 0, 14, 0, 7, 15, 53, 0, 81, 0, 110, 61, 32, 0, 562, 170, 62, 621, 560, 0, 1400, 0, 3387, 569, 199, 1515, 7734, 0, 339, 1486, 13374, 0, 11926, 0, 8033, 27164, 913, 0, 85326, 15947, 47588, 8294, 25430, 0, 174779, 39748, 169009

Offset: 0

Gus Wiseman, Jan 18 2025

nonn

Also the number of integer partitions of n containing 1 whose product is a multiple of n. Without requiring a 1 we get A057568.

			The a(5) = 1 through a(11) = 14 partitions:
  (3,2)  .  (4,2,1)    (3,3,2)    (5,4)      .  (8,3)
            (2,2,2,1)  (3,3,1,1)  (5,2,2)       (4,4,3)
                                  (5,2,1,1)     (6,3,2)
                                                (6,4,1)
                                                (4,3,2,2)
                                                (4,3,3,1)
                                                (6,2,2,1)
                                                (3,2,2,2,2)
                                                (3,3,2,2,1)
                                                (4,3,2,1,1)
                                                (6,2,1,1,1)
                                                (3,2,2,2,1,1)
                                                (4,3,1,1,1,1)
                                                (3,2,2,1,1,1,1)

For n instead of n+1 we have A057568 (strict A379733), ranks A326149.
These partitions are ranked by A380217 = A379319/2 = (even case of A326149)/2.
The case of equality is A380218, see also A028422 = A001055 - 1 (ranks A325041).
A000041 counts integer partitions, strict A000009.
A379666 counts partitions by sum and product.
A380219 counts partitions of n whose product is a proper multiple of n, ranks A380216.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568, ranks A326149
- 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, ranks A379722, see A111133
Cf. A003963, A036844, A069016, A318950, A319000, A319916, A324851, A325042, A326150, A326152, A326156, A379671, A379678, A379734.

Table[Length[Select[IntegerPartitions[n],Divisible[Times@@#,n+1]&]],{n,0,30}]

a(n) = my(nb=0); forpart(p=n, if (!(vecprod(Vec(p)) % (n+1)), nb++)); nb; \\ Michel Marcus, Jan 21 2025

cp's OEIS Frontend

A331384 Numbers whose sum of prime factors is equal to their product of prime indices.

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A379734 Number of integer partitions of n into parts > 1 whose product is a multiple of n.

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A340828 Number of strict integer partitions of n whose maximum part is a multiple of their length.

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A340830 Number of strict integer partitions of n such that every part is a multiple of the number of parts.

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A379319 Even numbers whose product of prime indices is a multiple of their sum of prime indices.

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A379735 Number of strict integer partitions of n into parts > 1 whose product is a multiple of n.

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A331380 Numbers whose sum of prime factors is divisible by their sum of prime indices.

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A340827 Number of strict integer partitions of n into divisors of n whose length also divides n.

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A331382 Numbers whose sum of prime factors is divisible by their product of prime indices.

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