A301988 -id:A301988 - cp's OEIS frontend

A301987 Heinz numbers of integer partitions whose product is equal to their sum.

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 84, 89, 97, 101, 103, 107, 108, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 200, 211, 223, 227, 229, 233, 239, 241, 251

Offset: 1

Gus Wiseman, Mar 30 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of reversed integer partitions begins: (1), (2), (3), (4), (2 2), (5), (6), (7), (8), (9), (10), (1 2 3), (11), (12), (13), (14), (15), (16), (17), (18), (19), (20), (21), (22), (23), (1 1 2 4), (24), (25), (26), (27), (28), (1 1 2 2 2), (29), (30).

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

Cf. A001055, A002865, A003963, A056239, A276024, A284640, A296150, A299701, A301957, A301988.

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..300])[];  # Alois P. Heinz, Mar 27 2019

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[300],Total[primeMS[#]]===Times@@primeMS[#]&]

A340387 Numbers whose sum of prime indices is twice their number, counted with multiplicity in both cases.

Original entry on oeis.org

1, 3, 9, 10, 27, 28, 30, 81, 84, 88, 90, 100, 208, 243, 252, 264, 270, 280, 300, 544, 624, 729, 756, 784, 792, 810, 840, 880, 900, 1000, 1216, 1632, 1872, 2080, 2187, 2268, 2352, 2376, 2430, 2464, 2520, 2640, 2700, 2800, 2944, 3000, 3648, 4896, 5440, 5616

Offset: 1

Gus Wiseman, Jan 09 2021

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.

Also Heinz numbers of integer partitions whose sum is twice their length, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). Like partitions in general (A000041), these are also counted by A000041.

			The sequence of terms together with their prime indices begins:
      1: {}
      3: {2}
      9: {2,2}
     10: {1,3}
     27: {2,2,2}
     28: {1,1,4}
     30: {1,2,3}
     81: {2,2,2,2}
     84: {1,1,2,4}
     88: {1,1,1,5}
     90: {1,2,2,3}
    100: {1,1,3,3}
    208: {1,1,1,1,6}
    243: {2,2,2,2,2}
    252: {1,1,2,2,4}

Partitions of 2n into n parts are counted by A000041.
The number of prime indices alone is A001222.
The sum of prime indices alone is A056239.
Allowing sum to be any multiple of length gives A067538, ranked by A316413.
A000569 counts graphical partitions, ranked by A320922.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A301987 lists numbers whose sum of prime indices equals their product, with nonprime case A301988.
Cf. A000720, A001221, A001414, A006125, A006129, A112798, A316428, A320911, A325037, A325044, A330950, A331385, A331416.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Total[primeMS[#]]==2*PrimeOmega[#]&]

All terms satisfy A056239(a(n)) = 2*A001222(a(n)).

A344415 Numbers whose greatest prime index is half their sum of prime indices.

Original entry on oeis.org

4, 9, 12, 25, 30, 40, 49, 63, 70, 84, 112, 121, 154, 165, 169, 198, 220, 264, 273, 286, 289, 325, 351, 352, 361, 364, 390, 442, 468, 520, 529, 561, 595, 624, 646, 714, 741, 748, 765, 832, 841, 850, 874, 918, 931, 952, 961, 988, 1020, 1045, 1173, 1197, 1224

Offset: 1

Gus Wiseman, May 19 2021

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:
       4: {1,1}           198: {1,2,2,5}
       9: {2,2}           220: {1,1,3,5}
      12: {1,1,2}         264: {1,1,1,2,5}
      25: {3,3}           273: {2,4,6}
      30: {1,2,3}         286: {1,5,6}
      40: {1,1,1,3}       289: {7,7}
      49: {4,4}           325: {3,3,6}
      63: {2,2,4}         351: {2,2,2,6}
      70: {1,3,4}         352: {1,1,1,1,1,5}
      84: {1,1,2,4}       361: {8,8}
     112: {1,1,1,1,4}     364: {1,1,4,6}
     121: {5,5}           390: {1,2,3,6}
     154: {1,4,5}         442: {1,6,7}
     165: {2,3,5}         468: {1,1,2,2,6}
     169: {6,6}           520: {1,1,1,3,6}

The partitions with these Heinz numbers are counted by A035363.
The conjugate version is A340387.
This sequence is the case of equality in A344414 and A344416.
A001222 counts prime factors with multiplicity.
A025065 counts palindromic partitions, ranked by A265640.
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.
A322109 ranks partitions of n with no part > n/2, counted by A110618.
A334201 adds up all prime indices except the greatest.
A344291 lists numbers m with A001222(m) <= A056239(m)/2, counted by A110618.
A344296 lists numbers m with A001222(m) >= A056239(m)/2, counted by A025065.
Cf. A000070, A001414, A209816, A301988, A316413, A316428, A320924, 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],Max[primeMS[#]]==Total[primeMS[#]]/2&]

A061395(a(n)) = A056239(a(n))/2.

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)).

A335405 Number of integer compositions of n with product n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 7, 1, 23, 11, 21, 1, 241, 1, 43, 73, 1092, 1, 1041, 1, 1339, 157, 111, 1, 23023, 137, 157, 1603, 3945, 1, 11599, 1, 153446, 421, 273, 601, 204586, 1, 343, 601, 206351, 1, 34789, 1, 16273, 25179, 507, 1, 5992730, 667, 33913, 1057, 27291, 1

Offset: 0

Gus Wiseman, Jun 06 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The compositions for n = 1, 4, 6, 8, 9, 10:
  (1)  (4)   (6)    (8)      (9)      (10)
       (22)  (123)  (1124)   (11133)  (11125)
             (132)  (1142)   (11313)  (11152)
             (213)  (1214)   (11331)  (11215)
             (231)  (1241)   (13113)  (11251)
             (312)  (1412)   (13131)  (11512)
             (321)  (1421)   (13311)  (11521)
                    (2114)   (31113)  (12115)
                    (2141)   (31131)  (12151)
                    (2411)   (31311)  (12511)
                    (4112)   (33111)  (15112)
                    (4121)            (15121)
                    (4211)            (15211)
                    (11222)           (21115)
                    (12122)           (21151)
                    (12212)           (21511)
                    (12221)           (25111)
                    (21122)           (51112)
                    (21212)           (51121)
                    (21221)           (51211)
                    (22112)           (52111)
                    (22121)
                    (22211)

Seiichi Manyama, Table of n, a(n) for n = 0..10000

The case of partitions is A001055.
Compositions are counted by A011782.
These compositions are ranked by A335404.
Cf. A003963, A096111, A124767, A233249, A272919, A301987, A301988, A333219, A333220, A331579.

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

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[#]]]&]

A335404 Numbers k such that the k-th composition in standard order (A066099) has the same product as sum.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 32, 37, 38, 41, 44, 50, 52, 64, 128, 139, 141, 142, 163, 171, 173, 174, 177, 181, 182, 184, 186, 197, 198, 209, 213, 214, 216, 218, 226, 232, 234, 256, 295, 307, 313, 316, 403, 409, 412, 457, 460, 484, 512, 535, 539, 541, 542, 647, 707, 737

Offset: 1

Gus Wiseman, Jun 06 2020

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence together with the corresponding compositions begins:
    1: (1)
    2: (2)
    4: (3)
    8: (4)
   10: (2,2)
   16: (5)
   32: (6)
   37: (3,2,1)
   38: (3,1,2)
   41: (2,3,1)
   44: (2,1,3)
   50: (1,3,2)
   52: (1,2,3)
   64: (7)
  128: (8)
  139: (4,2,1,1)
  141: (4,1,2,1)
  142: (4,1,1,2)
  163: (2,4,1,1)
  171: (2,2,2,1,1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

The lengths of standard compositions are given by A000120.
Sum of binary indices is A029931.
Sum of prime indices is A056239.
Sum of standard compositions is A070939.
Product of standard compositions is A124758.
Taking GCD instead of product gives A131577.
The version for prime indices is A301987.
The version for prime indices of nonprime numbers is A301988.
These compositions are counted by A335405.
Cf. A001055, A003963, A066099, A096111, A124767, A228351, A233249, A272919, A333219, A333220, A331579.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],Times@@stc[#]==Plus@@stc[#]&]

A124758(a(n)) = A070939(a(n)).

A380217 Numbers whose product of prime indices is a multiple of their sum of prime indices plus one.

Original entry on oeis.org

1, 15, 42, 54, 75, 77, 95, 99, 100, 132, 182, 195, 221, 234, 245, 253, 290, 312, 315, 329, 350, 357, 405, 420, 423, 437, 450, 459, 476, 494, 510, 540, 555, 559, 560, 612, 627, 665, 715, 720, 740, 798, 816, 833, 854, 855, 858, 893, 897, 899, 979, 1026, 1064

Offset: 1

Gus Wiseman, Jan 18 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 75 are {2,3,3}, with product 18 and sum 8, and since 18 is a multiple of 8+1, 75 is in the sequence.
The terms together with their prime indices begin:
     1: {}
    15: {2,3}
    42: {1,2,4}
    54: {1,2,2,2}
    75: {2,3,3}
    77: {4,5}
    95: {3,8}
    99: {2,2,5}
   100: {1,1,3,3}
   132: {1,1,2,5}
   182: {1,4,6}
   195: {2,3,6}
   221: {6,7}
   234: {1,2,2,6}
   245: {3,4,4}

The case of equality is A325041, counted by A380218 = A028422 except n=3.
Without "plus one" we get A326149, counted by A057568, see A379733, A379734, A379735.
Double all terms to get A379319.
Partitions of this type are counted by A379320.
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. A069016, A301988, A318950, A319000, A324851, A379318, A379671, A379844, A379845.

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

vpind(n)=my(v=List(), f=factor(n)); for(i=1, #f~, for(j=1, f[i, 2], listput(v, primepi(f[i, 1])))); Vec(v); \\ A112798
isok(k) = my(vind = vpind(k)); (vecprod(vind) % (vecsum(vind)+1)) == 0; \\ Michel Marcus, Jan 21 2025

a(n) = A379319(n)/2.

A344413 Numbers n whose sum of prime indices A056239(n) is even and is at least twice the number of prime factors A001222(n).

Original entry on oeis.org

1, 3, 7, 9, 10, 13, 19, 21, 22, 25, 27, 28, 29, 30, 34, 37, 39, 43, 46, 49, 52, 53, 55, 57, 61, 62, 63, 66, 70, 71, 75, 76, 79, 81, 82, 84, 85, 87, 88, 89, 90, 91, 94, 100, 101, 102, 107, 111, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138

Offset: 1

Gus Wiseman, May 19 2021

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.

Also Heinz numbers of integer partitions of even numbers m with at most m/2 parts, counted by A209816 riffled with zeros, or A110618 with odd positions zeroed out.

			The sequence of terms together with their prime indices begins:
      1: {}          37: {12}        75: {2,3,3}
      3: {2}         39: {2,6}       76: {1,1,8}
      7: {4}         43: {14}        79: {22}
      9: {2,2}       46: {1,9}       81: {2,2,2,2}
     10: {1,3}       49: {4,4}       82: {1,13}
     13: {6}         52: {1,1,6}     84: {1,1,2,4}
     19: {8}         53: {16}        85: {3,7}
     21: {2,4}       55: {3,5}       87: {2,10}
     22: {1,5}       57: {2,8}       88: {1,1,1,5}
     25: {3,3}       61: {18}        89: {24}
     27: {2,2,2}     62: {1,11}      90: {1,2,2,3}
     28: {1,1,4}     63: {2,2,4}     91: {4,6}
     29: {10}        66: {1,2,5}     94: {1,15}
     30: {1,2,3}     70: {1,3,4}    100: {1,1,3,3}
     34: {1,7}       71: {20}       101: {26}
For example, 75 has 3 prime indices {2,3,3} with sum 8 >= 2*3, so 75 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

These are the Heinz numbers of partitions counted by A209816 and A110618.
A subset of A300061 (sum of prime indices is even).
The conjugate version appears to be A320924 (allowing odd weights: A322109).
The case of equality is A340387.
Allowing odd weights gives A344291.
The 5-smooth case is A344295, or A344293 allowing odd weights.
The opposite version allowing odd weights is A344296.
The conjugate opposite version allowing odd weights is A344414.
The case of equality in the conjugate case is A344415.
The conjugate opposite version is A344416, counted by A000070.
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.
A330950 counts partitions of n with Heinz number divisible by n.
A334201 adds up all prime indices except the greatest.
Cf. A001414, A025065, A067538, A301988, A316413, A316428, A325037, A325038, A325044, A338914, A344294, A344297.

filter:= proc(n) local F,a,t;
  F:= ifactors(n)[2];
  a:= add((numtheory:-pi(t[1])-2)*t[2],t=F);
  a::even and a >= 0
end proc:
select(filter, [$1..300]); # Robert Israel, Oct 10 2024

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

Members m of A300061 such that A056239(m) >= 2*A001222(m).

A380216 Numbers whose prime indices have (product)/(sum) equal to an integer > 1.

Original entry on oeis.org

49, 63, 65, 81, 125, 150, 154, 165, 169, 190, 198, 259, 273, 333, 351, 361, 364, 385, 390, 435, 442, 468, 481, 490, 495, 506, 525, 561, 580, 595, 609, 630, 658, 675, 700, 714, 741, 765, 781, 783, 810, 840, 841, 846, 874, 900, 918, 925, 931, 935, 952, 988

Offset: 1

Gus Wiseman, Jan 23 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 terms together with their prime indices begin:
   49: {4,4}
   63: {2,2,4}
   65: {3,6}
   81: {2,2,2,2}
  125: {3,3,3}
  150: {1,2,3,3}
  154: {1,4,5}
  165: {2,3,5}
  169: {6,6}
  190: {1,3,8}
  198: {1,2,2,5}
  259: {4,12}
  273: {2,4,6}
  333: {2,2,12}
  351: {2,2,2,6}
  361: {8,8}
  364: {1,1,4,6}
For example, 198 has prime indices {1,2,2,5}, and 20/10 is an integer > 1, so 198 is in the sequence.

The fraction A003963(n)/A056239(n) reduces to A326153(n)/A326154(n).
The non-proper version is A326149, superset of A326150.
Also a superset of A326151.
The squarefree case is A326158 without first term.
Partitions of this type are counted by A380219.
A379666 counts partitions by sum and product.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568 (strict A379733), ranks A326149, see A379735, A380217.
- 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. A000720, A001222, A028422, A036844, A112798, A301988, A319000, A324850, A324851, A326156, A379319, A379844.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],Divisible[Times@@prix[#],Total[prix[#]]]&&!SameQ[Times@@prix[#],Total[prix[#]]]&]

cp's OEIS Frontend

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A344415 Numbers whose greatest prime index is half their sum of prime indices.

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A335405 Number of integer compositions of n with product n.

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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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A335404 Numbers k such that the k-th composition in standard order (A066099) has the same product as sum.

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A380217 Numbers whose product of prime indices is a multiple of their sum of prime indices plus one.

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