A034957 Divide natural numbers in groups with prime(n) elements and add together.

1, 9, 35, 91, 242, 442, 833, 1273, 2024, 3306, 4464, 6586, 8897, 11137, 14288, 18762, 24190, 28670, 35778, 42813, 49275, 59329, 69056, 81702, 98067, 112110, 124836, 140919, 155325, 173568, 210312, 233835, 262903, 285923, 327949, 355001, 393285

Offset: 1

Author

Patrick De Geest, Oct 15 1998

Keywords

• nonn

Comments

Natural numbers starting from 0,1,2,3,...

Examples

			{0,1} #2 S=1;
{2,3,4} #3 S=9;
{5,6,7,8,9} #5 S=35;
{10,11,12,13,14,15,16} #7 S=91.
		

Links

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

Crossrefs

Cf. A006003, A027441, A034956.

Cf. A046992, A034958, A034959, A034960.

Programs

Mathematica

{1}~Join~Map[Abs@ Apply[Subtract, Map[PolygonalNumber, #]] &, Partition[Accumulate@ Prime@ Range@ 37 - 1, 2, 1]] (* Michael De Vlieger, Oct 06 2019 *)
Module[{nn=40,tprs},tprs=Total[Prime[Range[nn]]];Total/@TakeList[Range[0,tprs],Prime[Range[nn]]]] (* Harvey P. Dale, Apr 18 2025 *)

Python

from itertools import islice
from sympy import nextprime
def A034957_gen(): # generator of terms
    a, p = 0, 2
    while True:
        yield p*((a<<1)+p-1)>>1
        a, p = a+p, nextprime(p)
A034957_list = list(islice(A034957_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-1), n > 1.

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

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

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

a(n) = A034959(n)/2.

a(n) = A034956(n) - A000040(n).

(End)