A034960 Divide odd numbers into groups with prime(n) elements and add together.

4, 21, 75, 189, 495, 897, 1683, 2565, 4071, 6641, 8959, 13209, 17835, 22317, 28623, 37577, 48439, 57401, 71623, 85697, 98623, 118737, 138195, 163493, 196231, 224321, 249775, 281945, 310759, 347249, 420751, 467801, 525943, 571985, 656047

Offset: 1

Author

Patrick De Geest, Oct 15 1998

Keywords

• nonn

Examples

			{1,3} #2 S=4;
{5,7,9} #3 S=21;
{11,13,15,17,19} #5 S=75;
{21,23,25,27,29,31,33} #7 S=189.
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006003, A027441, A034959.

Cf. A007504.

Cf. A000040, A034956, A034957, A034958, A046992, A061802.

Programs

Maple

S:= n-> sum(ithprime(k), k=1..n): seq(S(n+1)^2-S(n)^2, n=0..40); # Gary Detlefs, Dec 20 2011

Mathematica

Accumulate[Join[{2}, ListConvolve[{1, 1}, #]]]*# & [Prime[Range[50]]] (* Paolo Xausa, Jun 23 2025 *)

PARI

a0(n) = vecsum(primes(n))^2 - vecsum(primes(n-1))^2; \\ Michel Marcus, Jun 16 2024

Python

from itertools import islice
from sympy import nextprime
def A034960_gen(): # generator of terms
    a, p = 0, 2
    while True:
        yield p*((a<<1)+p)
        a, p = a+p, nextprime(p)
A034960_list = list(islice(A034960_gen(),20)) # Chai Wah Wu, Mar 22 2023

Formula

From Hieronymus Fischer, Sep 26 2012: (Start)

a(n) = Sum_{k=A007504(n-1)+1..A007504(n)} (2*k-1).

a(n) = A007504(n)^2 - A007504(n-1)^2.

a(n) = 2*A034957(n) + A000040(n).

a(n) = 2*A034956(n) - A000040(n).

a(n) = A034959(n) + A000040(n). (End)

a(n) = A061802(n)*A000040(n). - Marco Zárate, May 12 2023