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A202523 Number of partitions of n into distinct parts having pairwise prime GCDs but no overall common factor. 1
1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 4, 0, 0, 0, 1, 0, 4, 0, 1, 0, 5, 0, 4, 0, 0, 0, 3, 0, 5, 0, 0, 0, 6, 0, 1, 1, 2, 0, 6, 0, 6, 1, 1, 0, 4, 0, 12, 0, 1, 1, 12, 1, 9, 0, 0, 1, 10, 0, 10, 0, 1, 2, 10, 1, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

31,11

LINKS

Alois P. Heinz, Table of n, a(n) for n = 31..251

EXAMPLE

a(31) = 1: [6,10,15] = [2*3,2*5,3*5].

a(37) = 1: [10,12,15] = [2*5,2*2*3,3*5].

a(41) = 2: [6,15,20], [6,14,21].

a(43) = 1: [10,15,18].

a(47) = 1: [12,14,21].

a(49) = 1: [10,15,24].

a(61) = 4: [6,22,33], [10,15,36], [6,15,40], [6,10,45].

MAPLE

w:=(m, h)-> mul(`if`(j[1]>=h, 1, j[1]^min(j[2], 2)), j=ifactors(m)[2]):

b:= proc(n, i, g, s) option remember; local j, ok, si;

if n=0 then `if`(g>1, 0, 1)

elif i<2 or member(1, s) then 0

else ok:= evalb(i<=n);

si:= map(x->w(x, i), s);

for j in s while ok do ok:= isprime(igcd(i, j)) od;

b(n, i-1, g, si) +`if`(ok,

b(n-i, i-1, igcd(i, g), si union {w(i, i)} ), 0)

fi

end:

a:= n-> b(n, n, 0, {}):

seq(a(n), n=31..100);

MATHEMATICA

w[m_, h_] := Product[If[j[[1]] >= h, 1, j[[1]]^Min[j[[2]], 2]], {j, FactorInteger[m]}];

b[n_, i_, g_, s_] := b[n, i, g, s] = Module[{j, ok, si}, Which[n == 0, If[g > 1, 0, 1], i < 2 || MemberQ[s, 1], 0, True, ok = i <= n; si = w[#, i]& /@ s; For[j = 1, j <= Length[s], j++, If[!ok, Break[]]; ok = PrimeQ[ GCD[i, s[[j]]]]]; b[n, i - 1, g, si] + If[ok, b[n - i, i - 1, GCD[i, g], si ~Union~ {w[i, i]}], 0]]];

a[n_] := b[n, n, 0, {}];

Table[a[n], {n, 31, 100}] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)

CROSSREFS

Cf. A202385, A202425, A200976, A018783.

Sequence in context: A289618 A161520 A070097 * A215935 A270573 A096271

Adjacent sequences: A202520 A202521 A202522 * A202524 A202525 A202526

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Dec 20 2011

STATUS

approved

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Last modified January 30 19:10 EST 2023. Contains 359947 sequences. (Running on oeis4.)