A325091 - OEIS

%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