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A258116 The Heinz numbers in increasing order of the partitions into distinct odd parts. 4
1, 2, 5, 10, 11, 17, 22, 23, 31, 34, 41, 46, 47, 55, 59, 62, 67, 73, 82, 83, 85, 94, 97, 103, 109, 110, 115, 118, 127, 134, 137, 146, 149, 155, 157, 166, 167, 170, 179, 187, 191, 194, 197, 205, 206, 211, 218, 227, 230, 233, 235, 241, 253, 254, 257, 269, 274, 277, 283, 295, 298, 307, 310, 313, 314, 331, 334, 335, 341, 347 (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

170 is in the sequence because it is the Heinz number of the partition [1,3,7]; indeed, (1st prime)*(3rd prime)*(7th prime) = 2*5*17 = 170.

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: DO := {}: for q to 350 do if `and`(nops(B(q)) = nops(convert(B(q), set)), map(type, convert(B(q), set), odd) = {true}) then DO := `union`(DO, {q}) else  end if end do: DO;

# 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[2]=1 and numtheory

        [pi](i[1])::odd, ifactors(k)[2]) 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], #[[2]] == 1 && OddQ[PrimePi[#[[1]]]]&], Break[]]]; k]; Join[{1}, Array[a, 100]] (* Jean-Fran├žois Alcover, Dec 10 2016 after Alois P. Heinz *)

CROSSREFS

Cf. A215366, A258117.

Sequence in context: A257031 A167799 A179871 * A324812 A032874 A240032

Adjacent sequences:  A258113 A258114 A258115 * A258117 A258118 A258119

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 September 22 16:41 EDT 2019. Contains 327311 sequences. (Running on oeis4.)