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

%I A325359 #12 Sep 30 2020 03:30:18
%S A325359 9,18,25,27,36,49,50,54,72,81,98,100,108,121,125,144,162,169,196,200,
%T A325359 216,242,243,250,288,289,324,338,343,361,392,400,432,484,486,500,529,
%U A325359 576,578,625,648,676,686,722,729,784,800,841,864,961,968,972,1000,1058
%N A325359 Numbers of the form p^y * 2^z where p is an odd prime, y >= 2, and z >= 0.
%C A325359 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.
%H A325359 Robert Israel, <a href="/A325359/b325359.txt">Table of n, a(n) for n = 1..10000</a>
%F A325359 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
%e A325359 The sequence of terms together with their prime indices begins:
%e A325359      9: {2,2}
%e A325359     18: {1,2,2}
%e A325359     25: {3,3}
%e A325359     27: {2,2,2}
%e A325359     36: {1,1,2,2}
%e A325359     49: {4,4}
%e A325359     50: {1,3,3}
%e A325359     54: {1,2,2,2}
%e A325359     72: {1,1,1,2,2}
%e A325359     81: {2,2,2,2}
%e A325359     98: {1,4,4}
%e A325359    100: {1,1,3,3}
%e A325359    108: {1,1,2,2,2}
%e A325359    121: {5,5}
%e A325359    125: {3,3,3}
%e A325359    144: {1,1,1,1,2,2}
%e A325359    162: {1,2,2,2,2}
%e A325359    169: {6,6}
%e A325359    196: {1,1,4,4}
%e A325359    200: {1,1,1,3,3}
%p A325359 N:= 1000: # to get terms <= N
%p A325359 P:= select(isprime, [seq(i,i=3..floor(sqrt(N)),2)]):
%p A325359 B:= map(proc(p) local y;  seq(p^y, y=2..floor(log[p](N))) end proc, P):
%p A325359 sort(map(proc(t) local z;  seq(2^z*t, z=0..ilog2(N/t)) end proc, B)); # _Robert Israel_, May 03 2019
%t A325359 Select[Range[1000],MatchQ[FactorInteger[2*#],{{2,_},{_?(#>2&),_?(#>1&)}}]&]
%Y A325359 Positions of 2's in A325355.
%Y A325359 Numbers n such that n does not belong to A093641 but A325351(n) does.
%Y A325359 Cf. A056239, A112798, A136141, A325366, A325389, A325394, A325395, A325396.
%K A325359 nonn
%O A325359 1,1
%A A325359 _Gus Wiseman_, May 02 2019