A340856 Squarefree numbers whose greatest prime index (A061395) is divisible by their number of prime factors (A001222).

2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 30, 31, 35, 37, 38, 39, 41, 43, 47, 53, 57, 58, 59, 61, 65, 67, 71, 73, 74, 78, 79, 83, 86, 87, 89, 91, 95, 97, 101, 103, 106, 107, 109, 111, 113, 122, 127, 129, 130, 131, 133, 137, 138, 139, 142, 143, 145

Offset: 1

Author

Gus Wiseman, Feb 05 2021

Keywords

• nonn

Comments

Also Heinz numbers of strict integer partitions whose greatest part is divisible by their number of parts. These partitions are counted by A340828.

Examples

			The sequence of terms together with their prime indices begins:
      2: {1}         31: {11}       71: {20}
      3: {2}         35: {3,4}      73: {21}
      5: {3}         37: {12}       74: {1,12}
      6: {1,2}       38: {1,8}      78: {1,2,6}
      7: {4}         39: {2,6}      79: {22}
     11: {5}         41: {13}       83: {23}
     13: {6}         43: {14}       86: {1,14}
     14: {1,4}       47: {15}       87: {2,10}
     17: {7}         53: {16}       89: {24}
     19: {8}         57: {2,8}      91: {4,6}
     21: {2,4}       58: {1,10}     95: {3,8}
     23: {9}         59: {17}       97: {25}
     26: {1,6}       61: {18}      101: {26}
     29: {10}        65: {3,6}     103: {27}
     30: {1,2,3}     67: {19}      106: {1,16}
		

Crossrefs

Note: Heinz number sequences are given in parentheses below.

The case of equality, and the reciprocal version, are both A002110.

The non-strict reciprocal version is A168659 (A340609).

The non-strict version is A168659 (A340610).

These are the Heinz numbers of partitions counted by A340828.

A001222 counts prime factors.

A006141 counts partitions whose length equals their minimum (A324522).

A056239 adds up the prime indices.

A061395 selects the maximum prime index.

A067538 counts partitions whose length divides their sum (A316413/A326836).

A112798 lists the prime indices of each positive integer.

A200750 counts partitions with length coprime to maximum (A340608).

A257541 gives the rank of the partition with Heinz number n.

A340830 counts strict partitions whose parts are multiples of the length.

Cf. A039900, A047993 (A106529), A064174, A143773 (A316428), A326837, A326849 (A326848), A340599, A340691.

Programs

Mathematica

Select[Range[2,100],SquareFreeQ[#]&&Divisible[PrimePi[FactorInteger[#][[-1,1]]],PrimeOmega[#]]&]