cp's OEIS Frontend

A325092 Heinz numbers of integer partitions of powers of 2 into powers of 2.

%I A325092 #8 Mar 28 2019 09:56:53
%S A325092 1,2,3,4,7,9,12,16,19,49,53,63,81,84,108,112,131,144,192,256,311,361,
%T A325092 719,931,1197,1539,1596,1619,2052,2128,2401,2736,2809,3087,3648,3671,
%U A325092 3969,4116,4864,5103,5292,5488,6561,6804,7056,8161,8748,9072,9408,11664,12096
%N A325092 Heinz numbers of integer partitions of powers of 2 into powers of 2.
%C A325092 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose prime indices are powers of 2 and whose sum of prime indices is also 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 A325092 1 is in the sequence because it has prime indices {} with sum 0 = 2^(-infinity).
%e A325092 The sequence of terms together with their prime indices begins:
%e A325092     1: {}
%e A325092     2: {1}
%e A325092     3: {2}
%e A325092     4: {1,1}
%e A325092     7: {4}
%e A325092     9: {2,2}
%e A325092    12: {1,1,2}
%e A325092    16: {1,1,1,1}
%e A325092    19: {8}
%e A325092    49: {4,4}
%e A325092    53: {16}
%e A325092    63: {2,2,4}
%e A325092    81: {2,2,2,2}
%e A325092    84: {1,1,2,4}
%e A325092   108: {1,1,2,2,2}
%e A325092   112: {1,1,1,1,4}
%e A325092   131: {32}
%e A325092   144: {1,1,1,1,2,2}
%e A325092   192: {1,1,1,1,1,1,2}
%e A325092   256: {1,1,1,1,1,1,1,1}
%e A325092   311: {64}
%p A325092 q:= n-> andmap(t-> t=2^ilog2(t), (l-> [l[], add(i, i=l)])(
%p A325092       map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]))):
%p A325092 select(q, [$1..15000])[];  # _Alois P. Heinz_, Mar 28 2019
%t A325092 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A325092 pow2Q[n_]:=IntegerQ[Log[2,n]];
%t A325092 Select[Range[1000],#==1||pow2Q[Total[primeMS[#]]]&&And@@pow2Q/@primeMS[#]&]
%Y A325092 Cf. A000720, A001222, A018819, A033844, A056239, A102378, A112798, A318400, A325091, A325093.
%K A325092 nonn
%O A325092 1,2
%A A325092 _Gus Wiseman_, Mar 27 2019