A350838 Heinz numbers of partitions with no adjacent parts of quotient 2.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset

1,2

Comments

Differs from A320340 in having 105: (4,3,2), 315: (4,3,2,2), 455: (6,4,3), etc.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with no adjacent prime indices of quotient 1/2.

Links

Table of n, a(n) for n=1..67.

Example

The terms and their prime indices begin:
      1: {}            19: {8}             38: {1,8}
      2: {1}           20: {1,1,3}         39: {2,6}
      3: {2}           22: {1,5}           40: {1,1,1,3}
      4: {1,1}         23: {9}             41: {13}
      5: {3}           25: {3,3}           43: {14}
      7: {4}           26: {1,6}           44: {1,1,5}
      8: {1,1,1}       27: {2,2,2}         45: {2,2,3}
      9: {2,2}         28: {1,1,4}         46: {1,9}
     10: {1,3}         29: {10}            47: {15}
     11: {5}           31: {11}            49: {4,4}
     13: {6}           32: {1,1,1,1,1}     50: {1,3,3}
     14: {1,4}         33: {2,5}           51: {2,7}
     15: {2,3}         34: {1,7}           52: {1,1,6}
     16: {1,1,1,1}     35: {3,4}           53: {16}
     17: {7}           37: {12}            55: {3,5}

Mathematica

primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], And@@Table[FreeQ[Divide@@@Partition[primeptn[#], 2, 1], 2], {i, 2, PrimeOmega[#]}]&]

Crossrefs

The version with quotients >= 2 is counted by A000929, sets A018819.

<= 2 is A342191, counted by A342094.

< 2 is counted by A342096, sets A045690.

> 2 is counted by A342098, sets A040039.

The sets version (subsets of prescribed maximum) is counted by A045691.

These partitions are counted by A350837.

The strict case is counted by A350840.

For differences instead of quotients we have A350842, strict A350844.

The complement is A350845, counted by A350846.

A000041 = integer partitions.

A000045 = sets containing n with all differences > 2.

A003114 = strict partitions with no successions, ranked by A325160.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

A116931 = partitions with no successions, ranked by A319630.

A116932 = partitions with differences != 1 or 2, strict A025157.

A323092 = double-free integer partitions, ranked by A320340.

A350839 = partitions with gaps and conjugate gaps, ranked by A350841.

Cf. A000302, A001105, A003000, A018819, A094537, A120641, A154402, A319613, A323093, A337135, A342097, A342095.

Sequence in context: A031943 A023800 A320340 * A364347 A228869 A088725

Adjacent sequences: A350835 A350836 A350837 * A350839 A350840 A350841

Keyword

nonn

Author

Gus Wiseman, Jan 18 2022

Status

approved