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A083339 a(n) is the number of distinct prime factors of n that occur in partitions into two primes when n is even and into three primes when n is odd.

%I A083339 #34 Feb 16 2025 08:32:49
%S A083339 0,0,0,1,0,1,0,0,1,1,0,0,0,1,2,0,0,0,0,0,2,1,0,0,1,1,1,0,0,0,0,0,2,1,
%T A083339 2,0,0,1,2,0,0,0,0,0,2,1,0,0,1,0,2,0,0,0,2,0,2,1,0,0,0,1,2,0,2,0,0,0,
%U A083339 2,0,0,0,0,1,2,0,2,0,0,0,1,1,0,0,2,1,2,0,0,0,2,0,2,1,2,0,0,0,2,0,0,0,0,0,3
%N A083339 a(n) is the number of distinct prime factors of n that occur in partitions into two primes when n is even and into three primes when n is odd.
%C A083339 Number of distinct prime factors of n that occur in prime-partitions confirming Goldbach's conjectures. (The original name of this sequence.)
%C A083339 Conjecture: Apart from k=2, A070826(k): 1, 3, 15, 105, 1155, 15015, 255255, gives the positions of records (each equal to k-1). This follows from the conjectured formula. - _Antti Karttunen_, Sep 14 2017
%H A083339 Antti Karttunen, <a href="/A083339/b083339.txt">Table of n, a(n) for n = 1..2048</a>
%H A083339 Antti Karttunen, <a href="/A083338/a083338.txt">Scheme-program for computing A083338 and A083339</a>
%H A083339 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachConjecture.html">Goldbach Conjecture.</a>
%H A083339 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%F A083339 If n is even, a(n) = A010051(n/2), if n is an odd prime, a(n) = 0, and for odd composites (conjecturally), a(n) = A001221(n). - _Antti Karttunen_, Sep 14 2017
%e A083339 For n = 14 = 2*7 = 3 + 11 = 7 + 7, only one factor of 14 occurs, thus a(14) = 1.
%e A083339 For n = 15 = 3*5 = 2 + 2 + 11 = 3 + 5 + 7 = 5 + 5+ 5, both factors of 15 occur, thus a(15) = 2.
%e A083339 For n = 105 = 3*5*7, with 35 different partitions into three primes, the partition 97 + 5 + 3 contains the prime factors 3 and 5, while the partition 79 + 19 + 7 contains 7, thus all three prime factors of 115 occur and a(115) = 3.
%e A083339 For n = 1155 = 3*5*7*11, among 891 different partitions into three primes, the following four partitions: 1129 + 23 + 3 = 1129 + 19 + 7 = 1109 + 41 + 5 = 1103 + 41 + 11 each have either 3, 5, 7 or 11 as one of their parts, thus a(1155) = 4.
%t A083339 Table[Count[Union@ Flatten@ Select[IntegerPartitions[n, {2 + Boole[OddQ@ n]}], AllTrue[#, PrimeQ] &], p_ /; Divisible[n, p]], {n, 105}] (* _Michael De Vlieger_, Sep 16 2017 *)
%Y A083339 Cf. A001221, A083338, A045917, A054860, A070826, A100484.
%K A083339 nonn
%O A083339 1,15
%A A083339 _Reinhard Zumkeller_, Apr 24 2003
%E A083339 Name edited and two further examples added by _Antti Karttunen_, Sep 14 2017