A034956 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

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)

EXAMPLE

{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.

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 *)

PROG

(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

CROSSREFS

Cf. A006003, A027441, A034957.