A325092 - OEIS

%I #8 Mar 28 2019 09:56:53

%S 1,2,3,4,7,9,12,16,19,49,53,63,81,84,108,112,131,144,192,256,311,361,

%T 719,931,1197,1539,1596,1619,2052,2128,2401,2736,2809,3087,3648,3671,

%U 3969,4116,4864,5103,5292,5488,6561,6804,7056,8161,8748,9072,9408,11664,12096

%N Heinz numbers of integer partitions of powers of 2 into 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 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 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    12: {1,1,2}

%e    16: {1,1,1,1}

%e    19: {8}

%e    49: {4,4}

%e    53: {16}

%e    63: {2,2,4}

%e    81: {2,2,2,2}

%e    84: {1,1,2,4}

%e   108: {1,1,2,2,2}

%e   112: {1,1,1,1,4}

%e   131: {32}

%e   144: {1,1,1,1,2,2}

%e   192: {1,1,1,1,1,1,2}

%e   256: {1,1,1,1,1,1,1,1}

%e   311: {64}

%p q:= n-> andmap(t-> t=2^ilog2(t), (l-> [l[], add(i, i=l)])(

%p       map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]))):

%p select(q, [$1..15000])[];  # _Alois P. Heinz_, Mar 28 2019

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t pow2Q[n_]:=IntegerQ[Log[2,n]];

%t Select[Range[1000],#==1||pow2Q[Total[primeMS[#]]]&&And@@pow2Q/@primeMS[#]&]

%Y Cf. A000720, A001222, A018819, A033844, A056239, A102378, A112798, A318400, A325091, A325093.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 27 2019