A325359 Numbers of the form p^y * 2^z where p is an odd prime, y >= 2, and z >= 0.

9, 18, 25, 27, 36, 49, 50, 54, 72, 81, 98, 100, 108, 121, 125, 144, 162, 169, 196, 200, 216, 242, 243, 250, 288, 289, 324, 338, 343, 361, 392, 400, 432, 484, 486, 500, 529, 576, 578, 625, 648, 676, 686, 722, 729, 784, 800, 841, 864, 961, 968, 972, 1000, 1058

Offset: 1

Author

Gus Wiseman, May 02 2019

Keywords

• nonn

Comments

Also Heinz numbers of integer partitions that are not hooks but whose augmented differences are hooks, where the Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k), and a hook is a partition of the form (n,1,1,...,1). The enumeration of these partitions by sum is given by A325459.

Examples

			The sequence of terms together with their prime indices begins:
     9: {2,2}
    18: {1,2,2}
    25: {3,3}
    27: {2,2,2}
    36: {1,1,2,2}
    49: {4,4}
    50: {1,3,3}
    54: {1,2,2,2}
    72: {1,1,1,2,2}
    81: {2,2,2,2}
    98: {1,4,4}
   100: {1,1,3,3}
   108: {1,1,2,2,2}
   121: {5,5}
   125: {3,3,3}
   144: {1,1,1,1,2,2}
   162: {1,2,2,2,2}
   169: {6,6}
   196: {1,1,4,4}
   200: {1,1,1,3,3}
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Positions of 2's in A325355.

Numbers n such that n does not belong to A093641 but A325351(n) does.

Cf. A056239, A112798, A136141, A325366, A325389, A325394, A325395, A325396.

Programs

Maple

N:= 1000: # to get terms <= N
P:= select(isprime, [seq(i,i=3..floor(sqrt(N)),2)]):
B:= map(proc(p) local y;  seq(p^y, y=2..floor(log[p](N))) end proc, P):
sort(map(proc(t) local z;  seq(2^z*t, z=0..ilog2(N/t)) end proc, B)); # Robert Israel, May 03 2019

Mathematica

Select[Range[1000],MatchQ[FactorInteger[2*#],{{2,},{?(#>2&),_?(#>1&)}}]&]

Formula

Sum_{n>=1} 1/a(n) = 2 * Sum_{p prime} 1/(p*(p-1)) - 1 = 2 * A136141 - 1 = 0.54631333809959025572... - Amiram Eldar, Sep 30 2020