A034956 Divide natural numbers in groups with prime(n) elements and add together.

3, 12, 40, 98, 253, 455, 850, 1292, 2047, 3335, 4495, 6623, 8938, 11180, 14335, 18815, 24249, 28731, 35845, 42884, 49348, 59408, 69139, 81791, 98164, 112211, 124939, 141026, 155434, 173681, 210439, 233966, 263040, 286062, 328098, 355152, 393442, 434558, 472777

Offset: 1

Author

Patrick De Geest, Oct 15 1998

Keywords

• nonn

Comments

Natural numbers starting from 1,2,3,4,...

Examples

			{1,2} #2 S=3;
{3,4,5} #3 S=12;
{6,7,8,9,10} #5 S=40;
{11,12,13,14,15,16,17} #7 S=98.
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A006003, A027441, A034957.

Cf. A046992, A034958, A034959, A034960.

Cf. A000040, A000217, A007504.

Programs

Maple

s:= proc(n) s(n):= `if`(n<1, 0, s(n-1)+ithprime(n)) end:
a:= n-> (t-> t(s(n))-t(s(n-1)))(i-> i*(i+1)/2):
seq(a(n), n=1..40);  # Alois P. Heinz, Mar 22 2023

Mathematica

Module[{nn=50,pr},pr=Prime[Range[nn]];Total/@TakeList[Range[ Total[ pr]], pr]](* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Oct 01 2017 *)

Python

from itertools import islice
from sympy import nextprime
def A034956_gen(): # generator of terms
    a, p = 0, 2
    while True:
        yield p*((a<<1)+p+1)>>1
        a, p = a+p, nextprime(p)
A034956_list = list(islice(A034956_gen(),20)) # Chai Wah Wu, Mar 22 2023

Formula

From Hieronymus Fischer, Sep 27 2012: (Start)

a(n) = Sum_{k=A007504(n-1)+1..A007504(n)} k, n > 1.

a(n) = (A007504(n) - A007504(n-1))*(A007504(n) + A007504(n-1) + 1)/2, n > 1.

a(n) = (A000217(A007504(n)) - A000217(A007504(n-1))), n > 0.

If we define A007504(0) := 0, then the formulas above are also true for n=1.

a(n) = (A034960(n) + A000040(n))/2.

a(n) = A034957(n) + A000040(n). (End)