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 A258117 The Heinz numbers in increasing order of the partitions into distinct even parts. 2
 1, 3, 7, 13, 19, 21, 29, 37, 39, 43, 53, 57, 61, 71, 79, 87, 89, 91, 101, 107, 111, 113, 129, 131, 133, 139, 151, 159, 163, 173, 181, 183, 193, 199, 203, 213, 223, 229, 237, 239, 247, 251, 259, 263, 267, 271, 273, 281, 293, 301, 303, 311, 317, 321, 337, 339, 349 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, the Heinz number of the partition [1, 1, 2, 4, 10] is 2*2*3*7*29 = 2436. In the Maple program the subprogram B yields the partition with Heinz number n. More terms are obtained if one replaces the 350 in the Maple program by a larger number. REFERENCES G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass. 1976. G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, Cambridge. LINKS Alois P. Heinz, Table of n, a(n) for n = 1..10000 EXAMPLE 213 is in the sequence because it is the Heinz number of the partition [2,20]; indeed, (2nd prime)*(20th prime) = 3*71 = 213. MAPLE with(numtheory): B := proc (n) local pf: pf := op(2, ifactors(n)): [seq(seq(pi(op(1, op(i, pf))), j = 1 .. op(2, op(i, pf))), i = 1 .. nops(pf))] end proc: DE := {}: for q to 350 do if `and`(nops(B(q)) = nops(convert(B(q), set)), map(type, convert(B(q), set), even) = {true}) then DE := `union`(DE, {q}) else  end if end do: DE; # second Maple program: a:= proc(n) option remember; local k;       for k from 1+`if`(n=1, 0, a(n-1)) do         if not false in map(i-> i=1 and numtheory         [pi](i)::even, ifactors(k)) then break fi       od; k     end: seq(a(n), n=1..100);  # Alois P. Heinz, May 10 2016 MATHEMATICA a[n_] := a[n] = Module[{k}, For[k = 1 + If[n == 1, 0, a[n - 1]], True, k++, If[AllTrue[FactorInteger[k], #[] == 1 && EvenQ[PrimePi[#[]]]&], Break[]]]; k]; Array[a, 100] (* Jean-François Alcover, Dec 12 2016 after Alois P. Heinz *) CROSSREFS Cf. A215366, A258116. Sequence in context: A310258 A310259 A118667 * A034017 A034021 A216516 Adjacent sequences:  A258114 A258115 A258116 * A258118 A258119 A258120 KEYWORD nonn AUTHOR Emeric Deutsch, May 20 2015 EXTENSIONS a(1)=1 inserted by Alois P. Heinz, May 10 2016 STATUS approved

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Last modified November 20 17:09 EST 2019. Contains 329337 sequences. (Running on oeis4.)