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 A325093 Heinz numbers of integer partitions into distinct powers of 2. 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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. LINKS Robert Israel, Table of n, a(n) for n = 1..10000 EXAMPLE 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} 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]]]&] PROG (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 CROSSREFS Cf. A000720, A001222, A018819, A033844, A056239, A102378, A112798, A318400, A325091, A325092. Sequence in context: A000837 A200144 A056498 * A018652 A125686 A297413 Adjacent sequences: A325090 A325091 A325092 * A325094 A325095 A325096 KEYWORD nonn AUTHOR Gus Wiseman, Mar 27 2019 STATUS approved

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