A020696 Let a,b,c,...k be all divisors of n; a(n) = (a+1)*(b+1)*...*(k+1).

2, 6, 8, 30, 12, 168, 16, 270, 80, 396, 24, 10920, 28, 720, 768, 4590, 36, 31920, 40, 41580, 1408, 1656, 48, 2457000, 312, 2268, 2240, 104400, 60, 5499648, 64, 151470, 3264, 3780, 3456, 76767600, 76, 4680, 4480, 15343020, 84, 19071360, 88, 372600, 353280, 6768

Offset: 1

Author

Amarnath Murthy, Jun 01 2003

Keywords

• nonn
• easy

Comments

Named "Vandiver's arithmetical function" by Sándor (2021), after the American mathematician Harry Schultz Vandiver (1882-1973). - Amiram Eldar, Jun 29 2022

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• József Sándor, On Vandiver's arithmetical function - I, Notes on Number Theory and Discrete Mathematics, Vol. 27, No. 3 (2021), pp. 29-38.
• Harry S. Vandiver, Problem 116, American Mathematical Monthly, Vol. 11, No. 2 (1904), pp. 38-39.

Crossrefs

Cf. A027750, A000005, A003959, A007955.

Cf. A057643 (LCM instead of product).

Cf. A299436 (exp).

Programs

Haskell

a020696 = product . map (+ 1) . a027750_row'
-- Reinhard Zumkeller, Mar 28 2015

Maple

a:= n-> mul(d+1, d=numtheory[divisors](n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Jun 30 2022~

Mathematica

Table[Times @@ (Divisors[n] + 1), {n, 43}] (* Ivan Neretin, May 27 2015 *)

PARI

a(n) = {d = divisors(n); return (prod(i=1, #d, d[i]+1));} \\ Michel Marcus, Jun 12 2013

Python

from math import prod
from sympy import divisors
def A020696(n): return prod(d+1 for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 30 2022

Formula

a(p) = 2(p+1), a(p^2) = 2(p+1)(p^2+1) for primes p.

a(n) = Product_{k = 1..A000005(n)} (A027750(n,k) + 1). - Reinhard Zumkeller, Mar 28 2015

a(n) = Product_{d|n} (d+1). - Amiram Eldar, Jun 29 2022

Extensions

Edited by Don Reble, Jun 05 2003