OFFSET
1,1
COMMENTS
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.
EXAMPLE
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.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Max[primeMS[#]]>=Total[primeMS[#]]/2&]
CROSSREFS
These partitions are counted by A025065 but are different from palindromic partitions, which have Heinz numbers A265640.
The case of equality in the conjugate version is A340387.
The case of equality is A344415.
The even-weight case is A344416.
A001222 counts prime factors with multiplicity.
A301987 lists numbers whose sum of prime indices equals their product.
A334201 adds up all prime indices except the greatest.
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 19 2021
STATUS
approved