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Let a,b,c,...k be all divisors of n; a(n) = (a+1)*(b+1)*...*(k+1).
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%I #39 Jun 30 2022 12:57:12

%S 2,6,8,30,12,168,16,270,80,396,24,10920,28,720,768,4590,36,31920,40,

%T 41580,1408,1656,48,2457000,312,2268,2240,104400,60,5499648,64,151470,

%U 3264,3780,3456,76767600,76,4680,4480,15343020,84,19071360,88,372600,353280,6768

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

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

%H Reinhard Zumkeller, <a href="/A020696/b020696.txt">Table of n, a(n) for n = 1..10000</a>

%H József Sándor, <a href="https://nntdm.net/volume-27-2021/number-3/29-38/">On Vandiver's arithmetical function - I</a>, Notes on Number Theory and Discrete Mathematics, Vol. 27, No. 3 (2021), pp. 29-38.

%H Harry S. Vandiver, <a href="https://doi.org/10.2307/2967876">Problem 116</a>, American Mathematical Monthly, Vol. 11, No. 2 (1904), pp. 38-39.

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

%F a(n) = Product_{k = 1..A000005(n)} (A027750(n,k) + 1). - _Reinhard Zumkeller_, Mar 28 2015

%F a(n) = Product_{d|n} (d+1). - _Amiram Eldar_, Jun 29 2022

%p a:= n-> mul(d+1, d=numtheory[divisors](n)):

%p seq(a(n), n=1..50); # _Alois P. Heinz_, Jun 30 2022~

%t Table[Times @@ (Divisors[n] + 1), {n, 43}] (* _Ivan Neretin_, May 27 2015 *)

%o (PARI) a(n) = {d = divisors(n); return (prod(i=1, #d, d[i]+1));} \\ _Michel Marcus_, Jun 12 2013

%o (Haskell)

%o a020696 = product . map (+ 1) . a027750_row'

%o -- _Reinhard Zumkeller_, Mar 28 2015

%o (Python)

%o from math import prod

%o from sympy import divisors

%o def A020696(n): return prod(d+1 for d in divisors(n,generator=True)) # _Chai Wah Wu_, Jun 30 2022

%Y Cf. A027750, A000005, A003959, A007955.

%Y Cf. A057643 (LCM instead of product).

%Y Cf. A299436 (exp).

%K nonn,easy

%O 1,1

%A _Amarnath Murthy_, Jun 01 2003

%E Edited by _Don Reble_, Jun 05 2003