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A382817 a(n) = number of primes among the partial sums of row n of Pascal's triangle (A007318).

%I A382817 #21 Apr 13 2025 11:49:16
%S A382817 0,1,1,1,2,1,1,2,2,0,2,1,3,2,3,2,3,1,1,2,1,1,2,2,1,1,2,3,3,0,2,7,2,0,
%T A382817 0,1,1,0,0,0,2,0,1,1,1,0,1,3,1,0,1,1,1,1,1,1,5,3,3,2,3,2,3,3,10,0,1,0,
%U A382817 1,0,2,2,2,0,0,1,1,0,2,1,1,1,2,1,2,0
%N A382817 a(n) = number of primes among the partial sums of row n of Pascal's triangle (A007318).
%F A382817 a(n) = 0 <=> n in { A258483 }.
%e A382817 The numbers in A008949 (partial sums of Pascal's triangle) begin thus:
%e A382817   1
%e A382817   1    2
%e A382817   1    3     4
%e A382817   1    4     7     8
%e A382817   1    5    11    15    16
%e A382817   1    6    16    26    31    32
%e A382817   1    7    22    42    57    63    64
%e A382817 Row n=4 includes exactly 2 primes, so a(4) = 2.
%p A382817 a:= n-> nops(select(isprime, ListTools[PartialSums]
%p A382817             ([seq(binomial(n, k), k=0..n)]))):
%p A382817 seq(a(n), n=0..100);  # _Alois P. Heinz_, Apr 07 2025
%t A382817 t = Accumulate /@ Table[Binomial[n, i], {n, 0, 100}, {i, 0, n}]; (* A037955 *)
%t A382817 Map[PrimeQ, t]; Table[Count[m[[n]], True], {n, 1, 100}]
%o A382817 (PARI) a(n) = my(v=vector(n+1, k, binomial(n,k-1))); #select(isprime, vector(#v, k, sum(i=1, k, v[i]))); \\ _Michel Marcus_, Apr 13 2025
%Y A382817 Cf. A007318, A008949, A258483, A382816.
%K A382817 nonn
%O A382817 0,5
%A A382817 _Clark Kimberling_, Apr 07 2025