A034953 Triangular numbers (A000217) with prime indices.

3, 6, 15, 28, 66, 91, 153, 190, 276, 435, 496, 703, 861, 946, 1128, 1431, 1770, 1891, 2278, 2556, 2701, 3160, 3486, 4005, 4753, 5151, 5356, 5778, 5995, 6441, 8128, 8646, 9453, 9730, 11175, 11476, 12403, 13366, 14028, 15051, 16110, 16471, 18336, 18721, 19503

Offset: 1

Author

Patrick De Geest, Oct 15 1998

Keywords

• nonn
• easy

Comments

The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004

Given a rectangular prism with sides 1, p, p^2 for p = n-th prime (n > 1), the area of the six sides divided by the volume gives a remainder which is 4*a(n). - J. M. Bergot, Sep 12 2011

The infinite sum over the reciprocals is given by 2*A179119. - Wolfdieter Lang, Jul 10 2019

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Triangular Number.

Crossrefs

Cf. A000217, A034954, A034955, A011756, A179119, A195678, A307868.

Cf. A054269, A067076, A082749.

Programs

Haskell

a034953 n = a034953_list !! (n-1)
a034953_list = map a000217 a000040_list
-- Reinhard Zumkeller, Sep 23 2011

Maple

a:= n-> (p-> p*(p+1)/2)(ithprime(n)):
seq(a(n), n=1..65);  # Alois P. Heinz, Apr 20 2022

Mathematica

t[n_] := n(n + 1)/2; Table[t[Prime[n]], {n, 44}] (* Robert G. Wilson v, Aug 12 2004 *)
(#(# + 1))/2&/@Prime[Range[50]] (* Harvey P. Dale, Feb 27 2012 *)
With[{nn=200},Pick[Accumulate[Range[nn]],Table[If[PrimeQ[n],1,0],{n,nn}],1]] (* Harvey P. Dale, Mar 05 2023 *)

PARI

forprime(p=2,1e3,print1(binomial(p+1,2)", ")) \\ Charles R Greathouse IV, Jul 19 2011

PARI

apply(n->binomial(n+1,2),primes(100)) \\ Charles R Greathouse IV, Jun 04 2013

Formula

a(n) = A000217(A000040(n)). - Omar E. Pol, Jul 27 2009

a(n) = Sum_{k=1..prime(n)} k. - Wesley Ivan Hurt, Apr 27 2021

Product_{n>=1} (1 - 1/a(n)) = A307868. - Amiram Eldar, Nov 07 2022