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A036497 Number of partitions of n into distinct primes (counting 1 as a prime).

%I A036497 #35 Dec 16 2024 14:44:33
%S A036497 1,1,1,2,1,2,2,2,3,2,3,3,3,4,4,4,5,5,6,7,7,8,8,9,10,10,11,11,11,13,13,
%T A036497 15,16,16,18,18,20,22,22,24,25,26,29,30,32,33,34,37,39,41,44,45,47,51,
%U A036497 53,57,59,61,64,67,72,76,79,82,86,89,95,100,103,108,112,118
%N A036497 Number of partitions of n into distinct primes (counting 1 as a prime).
%C A036497 Honsberger shows that the primes-including-1 are a complete sequence and therefore all numbers in this sequence exceed zero. - _Ron Knott_, Aug 27 2016
%C A036497 Number of partitions of n into distinct noncomposite numbers. - _Omar E. Pol_, Dec 14 2024
%D A036497 Ross Honsberger, Mathematical Gems III, The Mathematical Association of America, 1985, pages 127-128.
%H A036497 Alois P. Heinz, <a href="/A036497/b036497.txt">Table of n, a(n) for n = 0..10000</a>
%F A036497 G.f.: (1 + x)*Product_{k>=1} (1 + x^prime(k)). - _Ilya Gutkovskiy_, Dec 31 2016
%e A036497 a(11) = 3 since 11 = 1+2+3+5=1+3+7 has 3 partitions of distinct primes-including-1. - _Ron Knott_, Aug 27 2016
%p A036497 s:= proc(n) option remember;
%p A036497       `if`(n<1, n+1, ithprime(n)+s(n-1))
%p A036497     end:
%p A036497 b:= proc(n, i) option remember; (p-> `if`(n=0, 1,
%p A036497       `if`(n>s(i), 0, b(n, i-1)+ `if`(p>n, 0,
%p A036497        b(n-p, i-1)))))(`if`(i<1, 1, ithprime(i)))
%p A036497     end:
%p A036497 a:= n-> b(n, numtheory[pi](n)):
%p A036497 seq(a(n), n=0..100);  # _Alois P. Heinz_, Aug 27 2016
%t A036497 myprime[ n_ ] := If[ n===0, 1, Prime[ n ] ]; ta1=Table[ Product[ 1+z^myprime[ k ], {k, 0, n} ]~CoefficientList~z, {n, 31, 32} ]; leveled=Count[ Take[ Last@ta1, Length@ta1[ [ -2 ] ] ]-ta1[ [ -2 ] ], 0 ]; Take[ Last@ta1, leveled ]
%t A036497 Table[Length@ DeleteCases[DeleteCases[IntegerPartitions@ n, {___, a_, ___} /; CompositeQ@ a], w_ /; MemberQ[Differences@ w, 0]], {n, 0, 60}] (* _Michael De Vlieger_, Aug 27 2016 *)
%Y A036497 Cf. A000586, A008578.
%K A036497 nonn
%O A036497 0,4
%A A036497 _Wouter Meeussen_, Dec 17 1998