A325093 Heinz numbers of integer partitions into distinct powers of 2.

1, 2, 3, 6, 7, 14, 19, 21, 38, 42, 53, 57, 106, 114, 131, 133, 159, 262, 266, 311, 318, 371, 393, 399, 622, 719, 742, 786, 798, 917, 933, 1007, 1113, 1438, 1619, 1834, 1866, 2014, 2157, 2177, 2226, 2489, 2751, 3021, 3238, 3671, 4314, 4354, 4857, 4978, 5033

Offset: 1

Author

Gus Wiseman, Mar 27 2019

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are squarefree numbers whose prime indices are powers 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.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    6: {1,2}
    7: {4}
   14: {1,4}
   19: {8}
   21: {2,4}
   38: {1,8}
   42: {1,2,4}
   53: {16}
   57: {2,8}
  106: {1,16}
  114: {1,2,8}
  131: {32}
  133: {4,8}
  159: {2,16}
  262: {1,32}
  266: {1,4,8}
  311: {64}
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000720, A001222, A018819, A033844, A056239, A102378, A112798, A318400, A325091, A325092.

Programs

Maple

P:= [seq(ithprime(2^i),i=0..20)]:f:= proc(S,N) option remember;
  if S = [] or S[1]>N then return {1} fi;
  procname(S[2..-1],N) union
    map(t -> S[1]*t, procname(S[2..-1], floor(N/S[1])))end proc:
sort(convert(f(P, P[20]),list));  # Robert Israel, Mar 28 2019

Mathematica

Select[Range[1000],SquareFreeQ[#]&&And@@IntegerQ/@Log[2,Cases[If[#==1,{},FactorInteger[#]],{p_,_}:>PrimePi[p]]]&]

PARI

isp2(q) = (q == 1) || (q == 2) || (ispower(q,,&p) && (p==2));
isok(n) = {if (issquarefree(n), my(f=factor(n)[,1]); for (k=1, #f, if (! isp2(primepi(f[k])), return (0));); return (1);); return (0);} \\ Michel Marcus, Mar 28 2019