A036497 - OEIS

%I #35 Dec 16 2024 14:44:33

%S 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 15,16,16,18,18,20,22,22,24,25,26,29,30,32,33,34,37,39,41,44,45,47,51,

%U 53,57,59,61,64,67,72,76,79,82,86,89,95,100,103,108,112,118

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

%C 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 Number of partitions of n into distinct noncomposite numbers. - _Omar E. Pol_, Dec 14 2024

%D Ross Honsberger, Mathematical Gems III, The Mathematical Association of America, 1985, pages 127-128.

%H Alois P. Heinz, <a href="/A036497/b036497.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: (1 + x)*Product_{k>=1} (1 + x^prime(k)). - _Ilya Gutkovskiy_, Dec 31 2016

%e 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 s:= proc(n) option remember;

%p       `if`(n<1, n+1, ithprime(n)+s(n-1))

%p     end:

%p b:= proc(n, i) option remember; (p-> `if`(n=0, 1,

%p       `if`(n>s(i), 0, b(n, i-1)+ `if`(p>n, 0,

%p        b(n-p, i-1)))))(`if`(i<1, 1, ithprime(i)))

%p     end:

%p a:= n-> b(n, numtheory[pi](n)):

%p seq(a(n), n=0..100);  # _Alois P. Heinz_, Aug 27 2016

%t 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 Table[Length@ DeleteCases[DeleteCases[IntegerPartitions@ n, {___, a_, ___} /; CompositeQ@ a], w_ /; MemberQ[Differences@ w, 0]], {n, 0, 60}] (* _Michael De Vlieger_, Aug 27 2016 *)

%Y Cf. A000586, A008578.

%K nonn

%O 0,4

%A _Wouter Meeussen_, Dec 17 1998