A317091
Positive integers whose prime multiplicities are weakly increasing and span an initial interval of positive integers.
Original entry on oeis.org
2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 101, 102
Offset: 1
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normalQ[m_]:=Union[m]==Range[Max[m]];
Select[Range[2,150],And[normalQ[FactorInteger[#][[All,2]]],OrderedQ[FactorInteger[#][[All,2]]]]&]
A335376
Heinz numbers of totally co-strong integer partitions.
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71
Offset: 1
The sequence of terms together with their prime indices begins:
1: {} 16: {1,1,1,1} 32: {1,1,1,1,1}
2: {1} 17: {7} 33: {2,5}
3: {2} 19: {8} 34: {1,7}
4: {1,1} 20: {1,1,3} 35: {3,4}
5: {3} 21: {2,4} 36: {1,1,2,2}
6: {1,2} 22: {1,5} 37: {12}
7: {4} 23: {9} 38: {1,8}
8: {1,1,1} 24: {1,1,1,2} 39: {2,6}
9: {2,2} 25: {3,3} 40: {1,1,1,3}
10: {1,3} 26: {1,6} 41: {13}
11: {5} 27: {2,2,2} 42: {1,2,4}
12: {1,1,2} 28: {1,1,4} 43: {14}
13: {6} 29: {10} 44: {1,1,5}
14: {1,4} 30: {1,2,3} 45: {2,2,3}
15: {2,3} 31: {11} 46: {1,9}
For example, 180 is the Heinz number of (3,2,2,1,1) which has run-lengths: (1,2,2) -> (1,2) -> (1,1) -> (2) -> (1). All of these are weakly increasing, so 180 is in the sequence.
Partitions with weakly increasing run-lengths are
A100883.
Totally strong partitions are counted by
A316496.
The version for reversed partitions is (also)
A316529.
These partitions are counted by
A332275.
The widely normal version is
A332293.
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primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
totcostrQ[q_]:=Or[Length[q]<=1,And[OrderedQ[Length/@Split[q]],totcostrQ[Length/@Split[q]]]];
Select[Range[100],totcostrQ[Reverse[primeMS[#]]]&]
A145108
Multiples of 4 that are primally tight and have strictly ascending powers.
Original entry on oeis.org
4, 8, 16, 32, 64, 108, 128, 256, 324, 512, 648, 972, 1024, 1944, 2048, 2916, 3888, 4096, 5832, 8192, 8748, 11664, 16384, 17496, 23328, 26244, 32768, 34992, 52488, 65536, 67500, 69984, 78732, 104976, 131072, 139968, 157464, 209952, 236196, 262144
Offset: 1
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