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

1,1

Comments

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

Links

Table of n, a(n) for n=1..61.

Example

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}

Mathematica

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

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.

Sequence in context: A362815 A358772 A003172 * A325100 A360007 A053329

Adjacent sequences: A340853 A340854 A340855 * A340857 A340858 A340859

Keyword

nonn

Author

Gus Wiseman, Feb 05 2021

Status

approved