A303975 -id:A303975 - cp's OEIS frontend

A330233 Least MM-numbers of multisets of multisets with a given number of distinct representatives (obtainable by vertex-permutations).

1, 35, 141, 1713, 28011, 355, 34567, 4045, 54849, 64615, 15265, 95363, 126841

Offset: 1

Gus Wiseman, Dec 09 2019

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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their corresponding multisets of multisets begins:
       1: {}
      35: {{2},{1,1}}
     141: {{1},{2,3}}
     355: {{2},{1,1,3}}
    1713: {{1},{2,3,4}}
    4045: {{2},{1,1,3,4}}
   15265: {{2},{1,4},{1,1,3}}
   28011: {{1},{2,3,4,5}}
   34567: {{1,2},{3,4,5}}
   54849: {{1},{2,3},{4,5}}
   64615: {{2},{1,1,3,4,5}}
   95363: {{2,3},{1,1,4,5}}
  126841: {{3},{1,2},{1,4,5}}

Sorted positions of first appearances in A330098.
The unsorted version is A330230.
The BII-number version is A330218.
MM-numbers of achiral multisets of multisets are A330232.
MM-numbers of fully-chiral multisets of multisets are A330236.
Cf. A001055, A003238, A007716, A056239, A112798, A302242, A303975, A322847, A330103, A330223, A330227, A330231.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Apply[Rule,Table[{p[[i]],i},{i,Length[p]}],{1}])],{p,Permutations[Union@@m]}]];
dv=Table[Length[graprms[primeMS/@primeMS[n]]],{n,1000}];
Table[Position[dv,i][[1,1]],{i,First/@Gather[dv]}]

A371127 Powers of 2 times powers > 1 of a prime-indexed prime number.

Original entry on oeis.org

3, 5, 6, 9, 10, 11, 12, 17, 18, 20, 22, 24, 25, 27, 31, 34, 36, 40, 41, 44, 48, 50, 54, 59, 62, 67, 68, 72, 80, 81, 82, 83, 88, 96, 100, 108, 109, 118, 121, 124, 125, 127, 134, 136, 144, 157, 160, 162, 164, 166, 176, 179, 191, 192, 200, 211, 216, 218, 236, 241

Offset: 1

Gus Wiseman, Mar 18 2024

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 terms together with their prime indices begin:
      3: {2}
      5: {3}
      6: {1,2}
      9: {2,2}
     10: {1,3}
     11: {5}
     12: {1,1,2}
     17: {7}
     18: {1,2,2}
     20: {1,1,3}
     22: {1,5}
     24: {1,1,1,2}
     25: {3,3}
     27: {2,2,2}
     31: {11}
     34: {1,7}
     36: {1,1,2,2}

Subset of A302540.
Subset of A336101 = powers of 2 times powers of primes.
Positions of 2's in A370820.
Counting prime factors instead of divisors gives A371287.
A000005 counts divisors.
A000961 lists powers of primes, A302596 of prime index.
A001221 counts distinct prime factors.
A003963 gives product of prime indices.
A027746 lists prime factors, indices A112798, length A001222.
A076610 lists products of primes of prime index.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
Cf. A000079, A005179, A007416, A303975, A319899, A355739, A370348, A370802, A371165-A371170, A371178.

Select[Range[100],Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]==2&]

A371178 Number of integer partitions of n containing all divisors of all parts.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 37, 48, 62, 80, 101, 127, 162, 202, 252, 312, 386, 475, 585, 713, 869, 1056, 1278, 1541, 1859, 2232, 2675, 3196, 3811, 4534, 5386, 6379, 7547, 8908, 10497, 12345, 14501, 16999, 19897, 23253, 27135, 31618, 36796, 42756

Offset: 0

Gus Wiseman, Mar 17 2024

nonn

The Heinz numbers of these partitions are given by A371177.

Also partitions such that the number of distinct parts is equal to the number of distinct divisors of parts.

			The partition (4,2,1,1) contains all distinct divisors {1,2,4}, so is counted under a(8).
The partition (4,4,3,2,2,2,1) contains all distinct divisors {1,2,3,4} so is counted under 4 + 4 + 3 + 2 + 2 + 2 + 1 = 18. - _David A. Corneth_, Mar 18 2024
The a(0) = 1 through a(8) = 12 partitions:
  ()  (1)  (11)  (21)   (31)    (221)    (51)      (331)      (71)
                 (111)  (211)   (311)    (321)     (421)      (521)
                        (1111)  (2111)   (2211)    (511)      (3221)
                                (11111)  (3111)    (2221)     (3311)
                                         (21111)   (3211)     (4211)
                                         (111111)  (22111)    (5111)
                                                   (31111)    (22211)
                                                   (211111)   (32111)
                                                   (1111111)  (221111)
                                                              (311111)
                                                              (2111111)
                                                              (11111111)

The LHS is represented by A001221, distinct case of A001222.
For partitions with no divisors of parts we have A305148, ranks A316476.
The RHS is represented by A370820, for prime factors A303975.
The strict case is A371128.
Counting all parts on the LHS gives A371130, ranks A370802.
The complement is counted by A371132.
For submultisets instead of distinct parts we have A371172, ranks A371165.
These partitions have ranks A371177.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
Cf. A000837, A003963, A239312, A285573, A305148, A319055, A355529, A370803, A370808, A370813, A371168, A371171, A371173.

Table[Length[Select[IntegerPartitions[n],SubsetQ[#,Union@@Divisors/@#]&]],{n,0,30}]

A371170 Positive integers with at most as many prime factors (A001222) as distinct divisors of prime indices (A370820).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 92

Offset: 1

Gus Wiseman, Mar 16 2024

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 prime indices of 105 are {2,3,4}, and there are 3 prime factors (3,5,7) and 4 distinct divisors of prime indices (1,2,3,4), so 105 is in the sequence.
The terms together with their prime indices begin:
     1: {}       22: {1,5}      42: {1,2,4}    63: {2,2,4}
     2: {1}      23: {9}        43: {14}       65: {3,6}
     3: {2}      25: {3,3}      45: {2,2,3}    66: {1,2,5}
     5: {3}      26: {1,6}      46: {1,9}      67: {19}
     6: {1,2}    28: {1,1,4}    47: {15}       69: {2,9}
     7: {4}      29: {10}       49: {4,4}      70: {1,3,4}
     9: {2,2}    30: {1,2,3}    51: {2,7}      71: {20}
    10: {1,3}    31: {11}       52: {1,1,6}    73: {21}
    11: {5}      33: {2,5}      53: {16}       74: {1,12}
    13: {6}      34: {1,7}      55: {3,5}      75: {2,3,3}
    14: {1,4}    35: {3,4}      57: {2,8}      76: {1,1,8}
    15: {2,3}    37: {12}       58: {1,10}     77: {4,5}
    17: {7}      38: {1,8}      59: {17}       78: {1,2,6}
    19: {8}      39: {2,6}      61: {18}       79: {22}
    21: {2,4}    41: {13}       62: {1,11}     82: {1,13}

The complement is A370348, counted by A371171.
The case of equality is A370802, counted by A371130, strict A371128.
The RHS is A370820, for prime factors instead of divisors A303975.
The strict version is A371168 counted by A371173.
The opposite version is A371169.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, indices A112798, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A003963, A319899, A355737, A355739, A355741, A370808, A371127.

Select[Range[100],PrimeOmega[#]<=Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A340652 Number of non-isomorphic twice-balanced multiset partitions of weight n.

Original entry on oeis.org

1, 1, 0, 2, 3, 6, 20, 65, 134, 482, 1562, 4974, 15466, 51768, 179055, 631737, 2216757, 7905325, 28768472, 106852116, 402255207, 1532029660, 5902839974, 23041880550, 91129833143, 364957188701, 1478719359501, 6058859894440, 25100003070184, 105123020009481, 445036528737301

Offset: 0

Gus Wiseman, Feb 07 2021

nonn

We define a multiset partition to be twice-balanced if all of the following are equal:
(1) the number of parts;
(2) the number of distinct vertices;
(3) the greatest size of a part.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 6 multiset partitions (empty column indicated by dot):
  {{1}}  .  {{1},{2,2}}  {{1,1},{2,2}}  {{1},{1},{2,3,3}}
            {{2},{1,2}}  {{1,2},{1,2}}  {{1},{2},{2,3,3}}
                         {{1,2},{2,2}}  {{1},{2},{3,3,3}}
                                        {{1},{3},{2,3,3}}
                                        {{2},{3},{1,2,3}}
                                        {{3},{3},{1,2,3}}

The co-balanced version is A319616.
The singly balanced version is A340600.
The cross-balanced version is A340651.
The version for factorizations is A340655.
A007716 counts non-isomorphic multiset partitions.
A007718 counts non-isomorphic connected multiset partitions.
A303975 counts distinct prime factors in prime indices.
A316980 counts non-isomorphic strict multiset partitions.
Other balance-related sequences:
- A047993 counts balanced partitions.
- A106529 lists balanced numbers.
- A340596 counts co-balanced factorizations.
- A340653 counts balanced factorizations.
- A340657/A340656 list numbers with/without a twice-balanced factorization.
Cf. A001055, A001221, A001222, A007717, A316983, A320663, A339888, A340597, A340599, A340654.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
G(m,n,k,y=1)={my(s=0); forpart(q=m, s+=permcount(q)*exp(sum(t=1, n, y^t*subst(x*Polrev(K(q, t, min(k,n\t))), x, x^t)/t, O(x*x^n)))); s/m!}
seq(n)={Vec(1 + sum(k=1,n, polcoef(G(k,n,k,y) - G(k-1,n,k,y) - G(k,n,k-1,y) + G(k-1,n,k-1,y), k, y)))} \\ Andrew Howroyd, Jan 15 2024

a(11) onwards from Andrew Howroyd, Jan 15 2024

A371169 Positive integers with at least as many prime factors (A001222) as distinct divisors of prime indices (A370820).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 12, 16, 18, 20, 22, 24, 25, 27, 28, 30, 32, 34, 36, 40, 42, 44, 45, 48, 50, 54, 56, 60, 62, 63, 64, 66, 68, 72, 75, 80, 81, 82, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 108, 110, 112, 118, 120, 121, 124, 125, 126, 128, 132, 134, 135

Offset: 1

Gus Wiseman, Mar 16 2024

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 terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    12: {1,1,2}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    22: {1,5}
    24: {1,1,1,2}
    25: {3,3}
    27: {2,2,2}
    28: {1,1,4}
    30: {1,2,3}
    32: {1,1,1,1,1}
    34: {1,7}
    36: {1,1,2,2}

The strict version is A370348 counted by A371171.
The case of equality is A370802, counted by A371130, strict A371128.
The RHS is A370820, for prime factors instead of divisors A303975.
The complement is A371168, counted by A371173.
The opposite version is A371170.
The version for prime factors instead of divisors on the RHS is A319899.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, indices A112798, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A003963, A355737, A355739, A355741, A370808, A370813, A370814, A371127.

Select[Range[100],PrimeOmega[#]>=Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A371166 Positive integers with fewer divisors (A000005) than distinct divisors of prime indices (A370820).

Original entry on oeis.org

7, 13, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 74, 79, 89, 91, 95, 97, 101, 103, 106, 107, 111, 113, 122, 131, 137, 139, 141, 142, 143, 145, 149, 151, 159, 161, 163, 167, 169, 173, 178, 181, 183, 185, 193, 197, 199, 203, 209, 213, 214, 215, 219, 221, 223, 226

Offset: 1

Gus Wiseman, Mar 14 2024

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 terms together with their prime indices begin:
     7: {4}       101: {26}      163: {38}      223: {48}
    13: {6}       103: {27}      167: {39}      226: {1,30}
    19: {8}       106: {1,16}    169: {6,6}     227: {49}
    23: {9}       107: {28}      173: {40}      229: {50}
    29: {10}      111: {2,12}    178: {1,24}    233: {51}
    37: {12}      113: {30}      181: {42}      239: {52}
    43: {14}      122: {1,18}    183: {2,18}    247: {6,8}
    47: {15}      131: {32}      185: {3,12}    251: {54}
    53: {16}      137: {33}      193: {44}      257: {55}
    61: {18}      139: {34}      197: {45}      259: {4,12}
    71: {20}      141: {2,15}    199: {46}      262: {1,32}
    73: {21}      142: {1,20}    203: {4,10}    263: {56}
    74: {1,12}    143: {5,6}     209: {5,8}     265: {3,16}
    79: {22}      145: {3,10}    213: {2,20}    267: {2,24}
    89: {24}      149: {35}      214: {1,28}    269: {57}
    91: {4,6}     151: {36}      215: {3,14}    271: {58}
    95: {3,8}     159: {2,16}    219: {2,21}    281: {60}
    97: {25}      161: {4,9}     221: {6,7}     293: {62}

The RHS is A370820, for prime factors instead of divisors A303975.
For (equal to) instead of (less than) we have A371165, counted by A371172.
For (greater than) instead of (less than) we have A371167.
For prime factors on the LHS we get A371168, counted by A371173.
Other equalities: A319899, A370802 (A371130), A371128, A371177 (A371178).
Other inequalities: A370348 (A371171), A371169, A371170.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, A112798 indices, length A001222.
A239312 counts divisor-choosable partitions, ranks A368110.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A370320 counts non-divisor-choosable partitions, ranks A355740.
A370814 counts divisor-choosable factorizations, complement A370813.
Cf. A000792, A003963, A355737, A355739, A355741, A368100, A370808, A371127.

Select[Range[100],Length[Divisors[#]] < Length[Union@@Divisors/@PrimePi/@First/@FactorInteger[#]]&]

A000005(a(n)) < A370820(a(n)).

A371131 Least number with exactly n distinct divisors of prime indices. Position of first appearance of n in A370820.

Original entry on oeis.org

1, 2, 3, 7, 13, 53, 37, 311, 89, 151, 223, 2045, 281, 3241, 1163, 827, 659, 9037, 1069, 17611, 1511, 4211, 28181, 122119, 2423, 10627, 88483, 6997, 7561, 98965, 5443, 88099, 6473, 95603, 309073, 50543, 10271, 192709, 508051, 438979, 14323, 305107, 26203

Offset: 0

Gus Wiseman, Mar 20 2024

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.
Every nonnegative integer belongs to A370820, so this sequence is infinite.
Are there any terms with more than two prime factors?

			The terms together with their prime indices begin:
       1: {}
       2: {1}
       3: {2}
       7: {4}
      13: {6}
      53: {16}
      37: {12}
     311: {64}
      89: {24}
     151: {36}
     223: {48}
    2045: {3,80}
     281: {60}
    3241: {4,90}
    1163: {192}
     827: {144}
     659: {120}
    9037: {4,210}
    1069: {180}
   17611: {5,252}

Counting prime factors instead of divisors (see A303975) gives A062447(>0).
The sorted version is A371181.
A000005 counts divisors.
A001221 counts distinct prime factors.
A003963 gives product of prime indices.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
Cf. A000720, A000792, A005179, A007416, A355739, A370348, A370802, A370808, A371130, A371165, A371177.

rnnm[q_]:=Max@@Select[Range[Min@@q,Max@@q],SubsetQ[q,Range[#]]&];
posfirsts[q_]:=Table[Position[q,n][[1,1]],{n,Min@@q,rnnm[q]}];
posfirsts[Table[Length[Union @@ Divisors/@PrimePi/@First/@If[n==1, {},FactorInteger[n]]],{n,1000}]]

f(n) = my(list=List(), f=factor(n)); for (i=1, #f~, fordiv(primepi(f[i,1]), d, listput(list, d))); #Set(list); \\ A370820
a(n) = my(k=1); while (f(k) != n, k++); k; \\ Michel Marcus, May 02 2024

A371132 Number of integer partitions of n with fewer distinct parts than distinct divisors of parts.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 6, 10, 14, 21, 28, 40, 53, 73, 96, 130, 170, 223, 288, 375, 480, 616, 780, 990, 1245, 1567, 1954, 2440, 3024, 3745, 4610, 5674, 6947, 8499, 10349, 12591, 15258, 18468, 22277, 26841, 32238, 38673, 46262, 55278, 65881, 78423, 93136, 110477

Offset: 0

Gus Wiseman, Mar 17 2024

nonn

The Heinz numbers of these partitions are given by A371179.

			The partition (4,3,1,1) has 3 distinct parts {1,3,4} and 4 distinct divisors of parts {1,2,3,4}, so is counted under a(9).
The a(0) = 0 through a(9) = 14 partitions:
  .  .  (2)  (3)  (4)   (5)   (6)    (7)     (8)      (9)
                  (22)  (32)  (33)   (43)    (44)     (54)
                        (41)  (42)   (52)    (53)     (63)
                              (222)  (61)    (62)     (72)
                              (411)  (322)   (332)    (81)
                                     (4111)  (422)    (333)
                                             (431)    (432)
                                             (611)    (441)
                                             (2222)   (522)
                                             (41111)  (621)
                                                      (3222)
                                                      (4311)
                                                      (6111)
                                                      (411111)

The LHS is represented by A001221, distinct case of A001222.
The RHS is represented by A370820, for prime factors A303975.
The complement counting all parts on the LHS is A371172, ranks A371165.
Counting all parts on the LHS gives A371173, ranks A371168.
The complement is counted by A371178, ranks A371177.
These partitions are ranked by A371179.
The strict case is A371180, complement A371128.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
Cf. A003963, A239312, A319055, A355740, A370802, A370803, A370808, A370813, A371130, A371171.

Table[Length[Select[IntegerPartitions[n],Length[Union[#]] < Length[Union@@Divisors/@#]&]],{n,0,30}]

A371167 Positive integers with more divisors (A000005) than distinct divisors of prime indices (A370820).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 34, 36, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 60, 62, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 82, 84, 85, 88, 90, 92, 93, 96, 98, 99, 100, 102, 104, 105, 108, 110

Offset: 1

Gus Wiseman, Mar 14 2024

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 prime indices of 814 are {1,5,12}, and there are 8 divisors (1,2,11,22,37,74,407,814) and 7 distinct divisors of prime indices (1,2,3,4,5,6,12), so 814 is in the sequence.
The prime indices of 1859 are {5,6,6}, and there are 6 divisors (1,11,13,143,169,1859) and 5 distinct divisors of prime indices (1,2,3,5,6), so 1859 is in the sequence.
The terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
     9: {2,2}
    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}
    27: {2,2,2}
    28: {1,1,4}
    30: {1,2,3}

For prime factors on the LHS we have A370348, counted by A371171.
The RHS is A370820, for prime factors instead of divisors A303975.
For (equal to) instead of (greater than) we get A371165, counted by A371172.
For (less than) instead of (greater than) we get A371166.
Other equalities: A319899, A370802 (A371130), A371128, A371177 (A371178).
Other inequalities: A371168 (A371173), A371169, A371170.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, A112798 indices, length A001222.
A239312 counts divisor-choosable partitions, ranks A368110.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A370320 counts non-divisor-choosable partitions, ranks A355740.
A370814 counts divisor-choosable factorizations, complement A370813.
Cf. A000792, A003963, A355529, A355739, A355741, A368100, A370808, A371127.

Select[Range[100],Length[Divisors[#]]>Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A000005(a(n)) > A370820(a(n)).

cp's OEIS Frontend

A330233 Least MM-numbers of multisets of multisets with a given number of distinct representatives (obtainable by vertex-permutations).

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A371127 Powers of 2 times powers > 1 of a prime-indexed prime number.

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A371178 Number of integer partitions of n containing all divisors of all parts.

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A371170 Positive integers with at most as many prime factors (A001222) as distinct divisors of prime indices (A370820).

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A340652 Number of non-isomorphic twice-balanced multiset partitions of weight n.

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Extensions

A371169 Positive integers with at least as many prime factors (A001222) as distinct divisors of prime indices (A370820).

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A371166 Positive integers with fewer divisors (A000005) than distinct divisors of prime indices (A370820).

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Formula

A371131 Least number with exactly n distinct divisors of prime indices. Position of first appearance of n in A370820.

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A371132 Number of integer partitions of n with fewer distinct parts than distinct divisors of parts.

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