cp's OEIS Frontend

A317137 a(n) is the number of nonzero triangular numbers <= n-th prime.

%I A317137 #43 Aug 02 2018 08:40:35
%S A317137 1,2,2,3,4,4,5,5,6,7,7,8,8,8,9,9,10,10,11,11,11,12,12,12,13,13,13,14,
%T A317137 14,14,15,15,16,16,16,16,17,17,17,18,18,18,19,19,19,19,20,20,20,20,21,
%U A317137 21,21,21,22,22,22,22,23,23,23,23,24,24,24,24,25,25,25,25,26,26,26,26,27,27,27,27,27
%N A317137 a(n) is the number of nonzero triangular numbers <= n-th prime.
%C A317137 a(n) is also the number of peaks in the largest Dyck path of the symmetric representation of sigma of the n-th prime (see example and A237593).
%F A317137 a(n) = A003056(A000040(n)).
%F A317137 a(n) = A123387(n) - 1.
%e A317137 Illustration of a(6) = 4:
%e A317137 .
%e A317137 .      _ _ _ _ _ _ _ 7
%e A317137 .     |_ _ _ _ _ _ _|
%e A317137 .                   |
%e A317137 .                   |_ _ 0
%e A317137 .                       |_ 0
%e A317137 .                         |
%e A317137 .                         |_ _ _ 7
%e A317137 .                             | |
%e A317137 .                             | |
%e A317137 .                             | |
%e A317137 .                             | |
%e A317137                               | |
%e A317137                               | |
%e A317137                               |_|
%e A317137 .
%e A317137 For n = 6 the 6th prime is A000040(6) = 13. Then we have that the (13 - 1)th = 12th row of triangle A237593 is [7, 2, 2, 1, 1, 2, 2, 7] and the 13th row of the same triangle is [7, 3, 2, 1, 1, 2, 3, 7], so the diagram of the symmetric representation of sigma(13) is constructed in the first quadrant as shown above. Note that the diagram has two parts and the total number of cells is 7 + 7 = 14, equaling the sum of the divisors of 13 (sigma(13) = 1 + 13 = 14). We can see that the largest Dyck path of the diagram has four peaks, so a(6) = 4.
%e A317137 On the other hand there are four nonzero triangular numbers <= 13, they are 1, 3, 6 and 10, so a(6) = 4.
%o A317137 (PARI) a(n) = sum(k=1, prime(n), ispolygonal(k, 3)); \\ _Michel Marcus_, Aug 01 2018
%Y A317137 Cf. A000040, A000203, A000217, A003056, A123387, A196020, A236104, A235791, A237048, A237270, A237271, A237591, A237593, A245092, A262626, A280850, A296508.
%K A317137 nonn,easy
%O A317137 1,2
%A A317137 _Omar E. Pol_, Jul 22 2018