cp's OEIS Frontend

A355306 Number of partitions of n in which the number of prime parts is not equal to the number of nonprime parts.

%I A355306 #27 Jul 16 2022 17:16:42
%S A355306 0,1,2,2,4,7,8,13,19,25,38,48,65,91,120,153,209,264,343,443,563,713,
%T A355306 912,1133,1428,1789,2217,2746,3406,4178,5139,6296,7670,9344,11360,
%U A355306 13732,16612,20038,24078,28915,34660,41402,49439,58887,69983,83088,98476,116436,137589,162244,191018
%N A355306 Number of partitions of n in which the number of prime parts is not equal to the number of nonprime parts.
%H A355306 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A355306 a(n) = A000041(n) - A155515(n).
%F A355306 a(n) = A355158(n) + A355225(n).
%e A355306 For n = 6 the partitions of 6 in which the number of prime parts is not equal to the number of nonprime parts are [6], [3, 3], [2, 2, 2], [3, 2, 1], [4, 1, 1], [3, 1, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1], there are eight of these partitions so a(6) = 8.
%t A355306 Array[Count[IntegerPartitions[#], _?(#1 - #2 != #2 & @@ {Length[#], Count[#, _?PrimeQ]} &)] &, 51, 0] (* _Michael De Vlieger_, Jul 15 2022 *)
%o A355306 (Python)
%o A355306 from sympy import isprime
%o A355306 from sympy.utilities.iterables import partitions
%o A355306 def c(p): return 2*sum(p[i] for i in p if isprime(i)) != sum(p.values())
%o A355306 def a(n): return sum(1 for p in partitions(n) if c(p))
%o A355306 print([a(n) for n in range(51)]) # _Michael S. Branicky_, Jun 28 2022
%o A355306 (PARI) a(n) = my(nb=0); forpart(p=n, if (#select(x->!isprime(x), Vec(p)) != #p/2, nb++)); nb; \\ _Michel Marcus_, Jun 30 2022
%Y A355306 Cf. A000040, A000041, A000607, A002095, A002096, A018252, A155515, A355158, A355225.
%K A355306 nonn
%O A355306 0,3
%A A355306 _Omar E. Pol_, Jun 28 2022