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

%I A020696 #39 Jun 30 2022 12:57:12
%S A020696 2,6,8,30,12,168,16,270,80,396,24,10920,28,720,768,4590,36,31920,40,
%T A020696 41580,1408,1656,48,2457000,312,2268,2240,104400,60,5499648,64,151470,
%U A020696 3264,3780,3456,76767600,76,4680,4480,15343020,84,19071360,88,372600,353280,6768
%N A020696 Let a,b,c,...k be all divisors of n; a(n) = (a+1)*(b+1)*...*(k+1).
%C A020696 Named "Vandiver's arithmetical function" by Sándor (2021), after the American mathematician Harry Schultz Vandiver (1882-1973). - _Amiram Eldar_, Jun 29 2022
%H A020696 Reinhard Zumkeller, <a href="/A020696/b020696.txt">Table of n, a(n) for n = 1..10000</a>
%H A020696 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 A020696 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 A020696 a(p) = 2(p+1), a(p^2) = 2(p+1)(p^2+1) for primes p.
%F A020696 a(n) = Product_{k = 1..A000005(n)} (A027750(n,k) + 1). - _Reinhard Zumkeller_, Mar 28 2015
%F A020696 a(n) = Product_{d|n} (d+1). - _Amiram Eldar_, Jun 29 2022
%p A020696 a:= n-> mul(d+1, d=numtheory[divisors](n)):
%p A020696 seq(a(n), n=1..50);  # _Alois P. Heinz_, Jun 30 2022~
%t A020696 Table[Times @@ (Divisors[n] + 1), {n, 43}] (* _Ivan Neretin_, May 27 2015 *)
%o A020696 (PARI) a(n) = {d = divisors(n); return (prod(i=1, #d, d[i]+1));} \\ _Michel Marcus_, Jun 12 2013
%o A020696 (Haskell)
%o A020696 a020696 = product . map (+ 1) . a027750_row'
%o A020696 -- _Reinhard Zumkeller_, Mar 28 2015
%o A020696 (Python)
%o A020696 from math import prod
%o A020696 from sympy import divisors
%o A020696 def A020696(n): return prod(d+1 for d in divisors(n,generator=True)) # _Chai Wah Wu_, Jun 30 2022
%Y A020696 Cf. A027750, A000005, A003959, A007955.
%Y A020696 Cf. A057643 (LCM instead of product).
%Y A020696 Cf. A299436 (exp).
%K A020696 nonn,easy
%O A020696 1,1
%A A020696 _Amarnath Murthy_, Jun 01 2003
%E A020696 Edited by _Don Reble_, Jun 05 2003