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%I #29 Mar 27 2017 15:33:06
%S 1,0,0,0,0,1,0,1,1,1,1,0,1,1,1,1,2,1,2,2,2,1,2,4,3,2,3,1,2,6,3,8,4,2,
%T 5,3,5,10,4,3,5,13,4,14,7,4,7,18,9,12,9,6,9,25,11,10,10,9,14,34,10,38,
%U 16,11,24,14,13,49,20,18,12,60,25,66,31,17,28
%N Number of partitions of n into distinct primes except the prime factors of n.
%C a(n) = 0 for n in {1, 2, 3, 4, 6, 11}, a(n) = 1 for n in {0, 5, 7, 8, 9, 10, 12, 13, 14, 15, 17, 21, 27}, a(n) = 2 for n in {16, 18, 19, 20, 22, 25, 28, 33}, a(n) = 3 for n in {24, 26, 30, 35, 39}, a(n) = 4 for n in {23, 32, 38, 42, 45}. The smallest n such that a(n) = 0, 1, 2, ... is 1, 0, 16, 24, 23, 34, 29, 44, 31, 48, 37, 54, 49, 41, 43, ... . Missing values are in {15, 19, 21, 22, 26, 33, 35, 36, 37, 39, 42, ... }.
%H Alois P. Heinz, <a href="/A209402/b209402.txt">Table of n, a(n) for n = 0..5000</a>
%F a(p) = A000586(p)-1 for any prime p.
%e a(5) = 1: [2,3].
%e a(7) = 1: [2,5].
%e a(16) = 2: [3,13], [5,11].
%e a(23) = 4: [3,7,13], [2,3,5,13], [5,7,11], [2,3,7,11].
%e a(24) = 3: [5,19], [7,17], [11,13].
%e a(29) = 6: [3,7,19], [2,3,5,19], [5,7,17], [2,3,7,17], [5,11,13], [2,3,11,13].
%e a(34) = 5: [3,31], [5,29], [11,23], [3,5,7,19], [3,7,11,13].
%p with(numtheory):
%p a:= proc(n) option remember; local b, l, f;
%p b:= proc(h, j) option remember;
%p `if`(h=0, 1, `if`(j<1, 0,
%p `if`(l[j]>h, 0, b(h-l[j], j-1)) +b(h, j-1)))
%p end; forget(b);
%p f:= factorset(n);
%p l:= sort([({ithprime(i)$i=1..pi(n)} minus f)[]]);
%p b(n, nops(l))
%p end:
%p seq(a(n), n=0..300);
%t a[n_] := a[n] = Module[{b, l, f}, b[h_, j_] := b[h, j] = If[h == 0, 1, If[j < 1, 0, If[l[[j]] > h, 0, b[h - l[[j]], j-1]] + b[h, j-1]]]; f = FactorInteger[n][[All, 1]]; l = Sort[Union[Array[Prime, PrimePi[n]] ~Complement~ f]]; b[n, Length[l]]]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Mar 27 2017, translated from Maple *)
%Y Cf. A000586, A208614.
%K nonn,look
%O 0,17
%A _Alois P. Heinz_, Mar 22 2012
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