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A258117
The Heinz numbers in increasing order of the partitions into distinct even parts.
17
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
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
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[2]=1 and numtheory
[pi](i[1])::even, 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 && EvenQ[PrimePi[#[[1]]]]&], Break[]]]; k]; Array[a, 100] (* Jean-François Alcover, Dec 12 2016 after Alois P. Heinz *)
CROSSREFS
Sequence in context: A118667 A353357 A352140 * A034017 A034021 A216516
KEYWORD
nonn
AUTHOR
Emeric Deutsch, May 20 2015
EXTENSIONS
a(1)=1 inserted by Alois P. Heinz, May 10 2016
STATUS
approved