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%I #12 Mar 28 2019 09:23:15
%S 1,2,3,4,7,9,10,12,16,19,34,39,49,52,53,55,63,66,70,75,81,84,88,90,94,
%T 100,108,112,120,129,131,144,160,172,192,205,246,254,256,259,311,328,
%U 333,339,341,361,370,377,391,434,444,452,465,545,558,592,598,609,614
%N Heinz numbers of integer partitions of powers of 2.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose sum of prime indices is a power of 2. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C 1 is in the sequence because it has prime indices {} with sum 0 = 2^(-infinity).
%e The sequence of terms together with their prime indices begins:
%e 1: {}
%e 2: {1}
%e 3: {2}
%e 4: {1,1}
%e 7: {4}
%e 9: {2,2}
%e 10: {1,3}
%e 12: {1,1,2}
%e 16: {1,1,1,1}
%e 19: {8}
%e 34: {1,7}
%e 39: {2,6}
%e 49: {4,4}
%e 52: {1,1,6}
%e 53: {16}
%e 55: {3,5}
%e 63: {2,2,4}
%e 66: {1,2,5}
%e 70: {1,3,4}
%e 75: {2,3,3}
%e 81: {2,2,2,2}
%p q:= n-> (t-> t=2^ilog2(t))(add(numtheory[pi](i[1])*i[2], i=ifactors(n)[2])):
%p select(q, [$1..1000])[]; # _Alois P. Heinz_, Mar 28 2019
%t Select[Range[100],#==1||IntegerQ[Log[2,Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]]]&]
%Y Cf. A000720, A001222, A018819, A033844, A056239, A102378, A112798, A131577, A318400, A325092, A325093.
%K nonn
%O 1,2
%A _Gus Wiseman_, Mar 27 2019
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