A325327 Heinz numbers of multiples of triangular partitions, or finite arithmetic progressions with offset 0.

1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 133, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 210, 211, 223, 227, 229, 233, 239

Offset

1,2

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Also numbers of the form Product_{k = 1..b} prime(k * c) for some b >= 0 and c > 0.

The enumeration of these partitions by sum is given by A007862.

Links

Table of n, a(n) for n=1..59.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Example

The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   21: {2,4}
   23: {9}
   29: {10}
   30: {1,2,3}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   53: {16}

Mathematica

primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], SameQ@@Differences[Append[primeptn[#], 0]]&]

Crossrefs

Cf. A000961, A007294, A007862, A049988, A056239, A112798, A130091, A289509, A307824, A325328, A325367, A325390, A325407.

Sequence in context: A283599 A096530 A299157 * A003174 A238463 A352936

Adjacent sequences: A325324 A325325 A325326 * A325328 A325329 A325330

Keyword

nonn

Author

Gus Wiseman, Apr 23 2019

Status

approved