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A386515 a(n) is the largest number of distinct primes in a partition of prime(n) into primes.

%I A386515 #63 Sep 02 2025 17:38:10
%S A386515 1,1,2,2,2,3,4,4,4,4,5,5,6,6,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,9,10,10,
%T A386515 10,10,10,10,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,13,13,13,13,
%U A386515 13,13,13,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15
%N A386515 a(n) is the largest number of distinct primes in a partition of prime(n) into primes.
%C A386515 For each prime number prime(n) find all sums of smaller prime numbers which add up to this prime number. Among those sums find the largest number of distinct primes.
%F A386515 a(n) <= A321578(n). - _David A. Corneth_, Aug 22 2025
%e A386515 Examples of such partitions for n = 3..11:
%e A386515   prime(3) = 5 = 2 + 3 which gives a(3) = 2;
%e A386515   prime(4) = 7 = 2 + 5 which gives a(4) = 2;
%e A386515   prime(5) = 11 = 2 + 2 + 2 + 2 + 3 = 3 + 3 + 5 which gives a(5)=2;
%e A386515   prime(6) = 13 = 2 + 3 + 5 + 3 which gives a(6)=3;
%e A386515   prime(7) = 17 = 2 + 3 + 5 + 7 which gives a(7)=4;
%e A386515   prime(8) = 19 = 2 + 3 + 5 + 7 + 2 which gives a(8)=4;
%e A386515   prime(9) = 23 = 2 + 3 + 5 + 13 which gives a(9)=4;
%e A386515   prime(10) = 29 = 2 + 3 + 5 + 19 which gives a(10)=4;
%e A386515   prime(11) = 31 = 2 + 3 + 5 + 7 + 7 + 7 which gives a(11)=4.
%Y A386515 Cf. A000040, A229219, A321578.
%K A386515 nonn,new
%O A386515 1,3
%A A386515 _Alexander R. Povolotsky_, Aug 21 2025
%E A386515 More terms from _David A. Corneth_, Aug 22 2025