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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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Last modified October 20 22:44 EDT 2019. Contains 328291 sequences. (Running on oeis4.)