A323014 -id:A323014 - cp's OEIS frontend

A324208 Numbers with exactly eight distinct exponents in their prime factorization, or eight distinct parts in their prime signature.

25968760179275365452000000, 29023908435660702564000000, 30690352939143613716000000, 31435867585438600284000000, 33959147926744708668000000, 34300982696689921212000000, 36356264250985511632800000, 37151479873700163972000000, 38953140268913048178000000, 39267640824717421116000000

Offset: 1

David A. Corneth, Feb 17 2019

nonn

			29023908435660702564000000 = 2^8 * 3^7 * 5^6 * 7^5 * 11^4 * 13^3 * 17 * 19^2 is in the sequence as there are exactly 8 distinct exponents; 1 through 8.

Cf. A001221, A001222, A006939, A051270, A059404, A071625, A118914, A181819, A182855, A323014, A323022, A323024, A323025, A323056.

PARI
```
is(n) = #Set(factor(n)[, 2]) == 8
```

A325256 Number of normal multisets of size n whose adjusted frequency depth is the maximum for multisets of that size.

Original entry on oeis.org

1, 1, 1, 2, 3, 10, 12, 12, 44, 128, 228, 422, 968, 1750, 420, 2100

Offset: 0

Gus Wiseman, Apr 18 2019

A multiset is normal if its union is an initial interval of positive integers.

The adjusted frequency depth of a multiset is 0 if the multiset is empty, and otherwise it is one plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the multiset {1,1,2,2,3} has adjusted frequency depth 5 because we have {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}. The enumeration of integer partitions by adjusted frequency depth is given by A325280. The adjusted frequency depth of the integer partition with Heinz number n is A323014(n).

			The a(1) = 1 through a(7) = 12 multisets:
  {1}  {12}  {112}  {1123}  {11123}  {111123}  {1112234}
             {122}  {1223}  {11223}  {111234}  {1112334}
                    {1233}  {11233}  {112345}  {1112344}
                            {11234}  {122223}  {1122234}
                            {12223}  {122234}  {1123334}
                            {12233}  {122345}  {1123444}
                            {12234}  {123333}  {1222334}
                            {12333}  {123334}  {1222344}
                            {12334}  {123345}  {1223334}
                            {12344}  {123444}  {1223444}
                                     {123445}  {1233344}
                                     {123455}  {1233444}

Cf. A011784, A181819, A182857, A225486, A323014, A323023, A325238, A325254, A325258, A325277, A325278, A325280, A325282, A325283.

nn=10;
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
mfdm=Table[Max@@fdadj/@allnorm[n],{n,0,nn}];
Table[Length[Select[allnorm[n],fdadj[#]==mfdm[[n+1]]&]],{n,0,nn}]

cp's OEIS Frontend

A324208 Numbers with exactly eight distinct exponents in their prime factorization, or eight distinct parts in their prime signature.

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