A307895 Numbers whose prime exponents, starting from the largest prime factor through to the smallest, form an initial interval of positive integers.

1, 2, 3, 5, 7, 11, 12, 13, 17, 19, 20, 23, 28, 29, 31, 37, 41, 43, 44, 45, 47, 52, 53, 59, 61, 63, 67, 68, 71, 73, 76, 79, 83, 89, 92, 97, 99, 101, 103, 107, 109, 113, 116, 117, 124, 127, 131, 137, 139, 148, 149, 151, 153, 157, 163, 164, 167, 171, 172, 173

Offset

1,2

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose multiplicities, starting from the largest part through to the smallest, form an initial interval of positive integers. The enumeration of these partitions by sum is given by A179269.

Links

Table of n, a(n) for n=1..60.

Example

The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   11: {5}
   12: {1,1,2}
   13: {6}
   17: {7}
   19: {8}
   20: {1,1,3}
   23: {9}
   28: {1,1,4}
   29: {10}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   44: {1,1,5}
   45: {2,2,3}

Mathematica

Select[Range[100], Last/@If[#==1, {}, FactorInteger[#]]==Range[PrimeNu[#], 1, -1]&]

Crossrefs

Cf. A055932, A056239, A098859, A109298, A112798, A130091, A179269, A317090, A325326, A325337, A325460.

Sequence in context: A342119 A324515 A212127 * A028835 A028834 A070026

Adjacent sequences: A307892 A307893 A307894 * A307896 A307897 A307898

Keyword

nonn

Author

Gus Wiseman, May 04 2019

Status

approved