A325755 Numbers n divisible by their prime shadow A181819(n).

1, 2, 9, 12, 18, 36, 40, 60, 84, 112, 120, 125, 132, 156, 180, 204, 225, 228, 250, 252, 276, 280, 336, 348, 352, 360, 372, 396, 440, 441, 444, 450, 468, 492, 516, 520, 540, 560, 564, 600, 612, 636, 675, 680, 684, 708, 732, 760, 804, 828, 832, 840, 852, 876

Offset

1,2

Comments

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

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 containing their multiset of multiplicities as a submultiset (counted by A325702).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     9: {2,2}
    12: {1,1,2}
    18: {1,2,2}
    36: {1,1,2,2}
    40: {1,1,1,3}
    60: {1,1,2,3}
    84: {1,1,2,4}
   112: {1,1,1,1,4}
   120: {1,1,1,2,3}
   125: {3,3,3}
   132: {1,1,2,5}
   156: {1,1,2,6}
   180: {1,1,2,2,3}
   204: {1,1,2,7}
   225: {2,2,3,3}
   228: {1,1,2,8}
   250: {1,3,3,3}
   252: {1,1,2,2,4}

Mathematica

red[n_]:=If[n==1, 1, Times@@Prime/@Last/@FactorInteger[n]];
Select[Range[100], Divisible[#, red[#]]&]

Crossrefs

Cf. A181819, A182857, A290689, A290822, A323014, A324843, A325702, A325706, A325708, A325756.

Sequence in context: A031070 A350630 A225547 * A360453 A324570 A109297

Adjacent sequences: A325752 A325753 A325754 * A325756 A325757 A325758

Keyword

nonn

Author

Gus Wiseman, May 19 2019

Status

approved