A330950 -id:A330950 - 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[#]&]

A344414 Heinz numbers of integer partitions whose sum is at most twice their greatest part.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85

Offset: 1

Gus Wiseman, May 19 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
     2: {1}        20: {1,1,3}    39: {2,6}
     3: {2}        21: {2,4}      40: {1,1,1,3}
     4: {1,1}      22: {1,5}      41: {13}
     5: {3}        23: {9}        42: {1,2,4}
     6: {1,2}      25: {3,3}      43: {14}
     7: {4}        26: {1,6}      44: {1,1,5}
     9: {2,2}      28: {1,1,4}    46: {1,9}
    10: {1,3}      29: {10}       47: {15}
    11: {5}        30: {1,2,3}    49: {4,4}
    12: {1,1,2}    31: {11}       51: {2,7}
    13: {6}        33: {2,5}      52: {1,1,6}
    14: {1,4}      34: {1,7}      53: {16}
    15: {2,3}      35: {3,4}      55: {3,5}
    17: {7}        37: {12}       56: {1,1,1,4}
    19: {8}        38: {1,8}      57: {2,8}
For example, 56 has prime indices {1,1,1,4} and 7 <= 2*4, so 56 is in the sequence. On the other hand, 224 has prime indices {1,1,1,1,1,4} and 9 > 2*4, so 224 is not in the sequence.

These partitions are counted by A025065 but are different from palindromic partitions, which have Heinz numbers A265640.
The opposite even-weight version appears to be A320924, counted by A209816.
The opposite version appears to be A322109, counted by A110618.
The case of equality in the conjugate version is A340387.
The conjugate opposite version is A344291, counted by A110618.
The conjugate opposite 5-smooth case is A344293, counted by A266755.
The conjugate version is A344296, also counted by A025065.
The case of equality is A344415.
The even-weight case is A344416.
A001222 counts prime factors with multiplicity.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A058696 counts partitions of even numbers, ranked by A300061.
A301987 lists numbers whose sum of prime indices equals their product.
A334201 adds up all prime indices except the greatest.
Cf. A001414, A067538, A301988, A316413, A316428, A325037, A325038, A325044, A330950, A344294, A344297.

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

A056239(a(n)) <= 2*A061395(a(n)).

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).

A344416 Heinz numbers of integer partitions whose sum is even and is at most twice the greatest part.

Original entry on oeis.org

3, 4, 7, 9, 10, 12, 13, 19, 21, 22, 25, 28, 29, 30, 34, 37, 39, 40, 43, 46, 49, 52, 53, 55, 57, 61, 62, 63, 66, 70, 71, 76, 79, 82, 84, 85, 87, 88, 89, 91, 94, 101, 102, 107, 111, 112, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138, 139, 146

Offset: 1

Gus Wiseman, May 20 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

Also numbers m whose sum of prime indices A056239(m) is even and is at most twice the greatest prime index A061395(m).

			The sequence of terms together with their prime indices begins:
      3: {2}         37: {12}          71: {20}
      4: {1,1}       39: {2,6}         76: {1,1,8}
      7: {4}         40: {1,1,1,3}     79: {22}
      9: {2,2}       43: {14}          82: {1,13}
     10: {1,3}       46: {1,9}         84: {1,1,2,4}
     12: {1,1,2}     49: {4,4}         85: {3,7}
     13: {6}         52: {1,1,6}       87: {2,10}
     19: {8}         53: {16}          88: {1,1,1,5}
     21: {2,4}       55: {3,5}         89: {24}
     22: {1,5}       57: {2,8}         91: {4,6}
     25: {3,3}       61: {18}          94: {1,15}
     28: {1,1,4}     62: {1,11}       101: {26}
     29: {10}        63: {2,2,4}      102: {1,2,7}
     30: {1,2,3}     66: {1,2,5}      107: {28}
     34: {1,7}       70: {1,3,4}      111: {2,12}

These partitions are counted by A000070 = even-indexed terms of A025065.
The opposite version appears to be A320924, counted by A209816.
The opposite version with odd weights allowed appears to be A322109.
The conjugate opposite version allowing odds is A344291, counted by A110618.
The conjugate version is A344296, also counted by A025065.
The conjugate opposite version is A344413, counted by A209816.
Allowing odd weight gives A344414.
The case of equality is A344415, counted by A035363.
A001222 counts prime factors with multiplicity.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A058696 counts partitions of even numbers, ranked by A300061.
A265640 lists Heinz numbers of palindromic partitions.
A301987 lists numbers whose sum of prime indices equals their product.
A334201 adds up all prime indices except the greatest.
A340387 lists Heinz numbers of partitions whose sum is twice their length.
Cf. A001414, A074761, A316413, A316428, A325037, A325038, A325044, A330950, A344293, A344294, A344297.

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

Intersection of A300061 and A344414.

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[#]]&]

A331385 Irregular triangle read by rows where T(n,k) is the number of integer partitions y of n such that Sum_i prime(y_i) = n + k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 3, 1, 0, 0, 0, 2, 3, 1, 1, 0, 0, 0, 1, 4, 3, 1, 2, 0, 0, 0, 0, 2, 5, 3, 2, 2, 0, 1, 0, 0, 0, 0, 1, 4, 6, 3, 4, 2, 0, 2, 0, 0, 0, 0, 0, 2, 6, 6, 4, 6, 2, 1, 2, 0, 1, 0, 0, 0, 0, 0, 1, 4, 8, 6, 6, 7, 2, 4, 2, 0, 1, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Jan 17 2020

			Triangle begins:
  1
  0 1
  0 1 1
  0 0 2 1
  0 0 1 3 1
  0 0 0 2 3 1 1
  0 0 0 1 4 3 1 2
  0 0 0 0 2 5 3 2 2 0 1
  0 0 0 0 1 4 6 3 4 2 0 2
  0 0 0 0 0 2 6 6 4 6 2 1 2 0 1
  0 0 0 0 0 1 4 8 6 6 7 2 4 2 0 1 0 0 0 1
  0 0 0 0 0 0 2 6 9 7 9 7 3 7 2 1 1 0 0 0 2
Row n = 8 counts the following partitions (empty column not shown):
  (2222)  (332)    (44)      (41111)    (53)        (611)   (8)
          (422)    (431)     (311111)   (62)        (5111)  (71)
          (3221)   (3311)    (2111111)  (521)
          (22211)  (4211)               (11111111)
                   (32111)
                   (221111)
Column k = 5 counts the following partitions:
  (11111)  (411)    (43)     (332)    (3222)   (22222)
           (3111)   (331)    (422)    (22221)
           (21111)  (421)    (3221)
                    (3211)   (22211)
                    (22111)

Row lengths are A331418.
Row sums are A000041.
Column sums are A331387.
Shifting row n to the right n times gives A331416.
Partitions whose sum of primes is divisible by their sum are A331379.
Partitions whose product divides their sum of primes are A331381.
Partitions whose product equals their sum of primes are A331383.
Cf. A000040, A001414, A014689, A056239, A330950, A330953, A330954, A331378, A331415.

Table[Length[Select[IntegerPartitions[n],Total[Prime/@#]==m&]],{n,0,10},{m,n,Max@@Table[Total[Prime/@y],{y,IntegerPartitions[n]}]}]

A331387 Number of integer partitions whose sum of primes of parts equals their sum of parts plus n.

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 24, 34, 47, 64, 86, 113, 148, 191, 245, 310, 390, 486, 602, 740, 907, 1104, 1338, 1613, 1937, 2315, 2758, 3272, 3871, 4562, 5362, 6283, 7344, 8558, 9952, 11542, 13356, 15419, 17766, 20425, 23440, 26846, 30696, 35032, 39917, 45406

Offset: 0

Gus Wiseman, Jan 17 2020

nonn

Primes of parts means the prime counting function applied to the part sizes. Equivalently, a(n) is the number of integer partitions with part sizes in A014689(n) interpreted as a multiset. - Andrew Howroyd, Apr 17 2021

			The a(0) = 1 through a(5) = 16 partitions:
  ()  (1)  (3)   (4)    (33)    (43)
      (2)  (11)  (31)   (41)    (331)
           (21)  (32)   (42)    (332)
           (22)  (111)  (311)   (411)
                 (211)  (321)   (421)
                 (221)  (322)   (422)
                 (222)  (1111)  (3111)
                        (2111)  (3211)
                        (2211)  (3221)
                        (2221)  (3222)
                        (2222)  (11111)
                                (21111)
                                (22111)
                                (22211)
                                (22221)
                                (22222)
For example, the partition (3,2,2,1) is counted under n = 5 because it has sum of primes 5+3+3+2 = 13 and its sum of parts plus n is also 3+2+2+1+5 = 13.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column sums of A331385.
Partitions into primes are A000607.
Partitions whose sum of primes is divisible by their sum are A331379.
Partitions whose product divides their sum of primes are A331381.
Partitions whose product equals their sum of primes are A331383.
Cf. A000040, A001414, A014689, A056239, A330950, A330953, A330954, A331378, A331415, A331416, A331418.

Table[Sum[Length[Select[IntegerPartitions[k],Total[Prime/@#]==k+n&]],{k,0,2*n}],{n,0,10}]

seq(n)={my(m=1); while(prime(m)-m<=n, m++); Vec(1/prod(k=1, m, 1 - x^(prime(k)-k) + O(x*x^n)))} \\ Andrew Howroyd, Apr 16 2021

G.f.: 1/Product_{k>=1} 1 - x^(prime(k)-k). - Andrew Howroyd, Apr 16 2021

Terms a(31) and beyond from Andrew Howroyd, Apr 16 2021

cp's OEIS Frontend

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

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A344414 Heinz numbers of integer partitions whose sum is at most twice their greatest part.

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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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A344416 Heinz numbers of integer partitions whose sum is even and is at most twice the greatest part.

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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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A331385 Irregular triangle read by rows where T(n,k) is the number of integer partitions y of n such that Sum_i prime(y_i) = n + k.