A003959 If n = Product p(k)^e(k) then a(n) = Product (p(k)+1)^e(k), a(1) = 1.

1, 3, 4, 9, 6, 12, 8, 27, 16, 18, 12, 36, 14, 24, 24, 81, 18, 48, 20, 54, 32, 36, 24, 108, 36, 42, 64, 72, 30, 72, 32, 243, 48, 54, 48, 144, 38, 60, 56, 162, 42, 96, 44, 108, 96, 72, 48, 324, 64, 108, 72, 126, 54, 192, 72, 216, 80, 90, 60, 216, 62, 96, 128, 729, 84, 144, 68

Offset: 1

Author

Marc LeBrun

Keywords

• nonn
• easy
• nice
• mult

Comments

Completely multiplicative.

Sum of divisors of n with multiplicity. If n = p^m, the number of ways to make p^k as a divisor of n is C(m,k); and sum(C(m,k)*p^k) = (p+1)^k. The rest follows because the function is multiplicative. - Franklin T. Adams-Watters, Jan 25 2010

Links

• Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
• Index to divisibility sequences

Crossrefs

Apart from initial terms, same as A064478.

Cf. A003958, A027746, A036987, A061142, A163407.

Cf. A063441, A165824, A165825, A165826, A165827.

Cf. A165828, A165829, A165831, A167350, A166590.

Cf. A166642, A166643, A166644, A166645, A166646.

Cf. A166647, A166649, A166650, A166586, A165830.

Cf. A167351, A168065, A168066.

Programs

Haskell

a003959 1 = 1
a003959 n = product $ map (+ 1) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012

Maple

a:= n-> mul((i[1]+1)^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

Mathematica

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]]+1)^fi[[All, 2]])); a /@ Range[67] (* Jean-François Alcover, Apr 22 2011 *)

PARI

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X-p*X))[n]) /* Ralf Stephan */

Formula

Multiplicative with a(p^e) = (p+1)^e. - David W. Wilson, Aug 01 2001

Sum_{n>0} a(n)/n^s = Product_{p prime} 1/(1-p^(-s)-p^(1-s)) (conjectured). - Ralf Stephan, Jul 07 2013

This follows from the absolute convergence of the sum (compare with a(n) = n^2) and the Euler product for completely multiplicative functions. Convergence occurs for at least Re(s)>3. - Thomas Anton, Jul 15 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = A065488/2 = 1/(2*A005596) = 1.3370563627850107544802059152227440187511993141988459926... - Vaclav Kotesovec, Jul 17 2021

From Thomas Scheuerle, Jul 19 2021: (Start)

a(n) = gcd(A166642(n), A166643(n)).

a(n) = A166642(n)/A061142(n).

a(n) = A166643(n)/A165824(n).

a(n) = A166644(n)/A165825(n).

a(n) = A166645(n)/A165826(n).

a(n) = A166646(n)/A165827(n).

a(n) = A166647(n)/A165828(n).

a(n) = A166649(n)/A165830(n).

a(n) = A166650(n)/A165831(n).

a(n) = A167351(n)/A166590(n). (End)

Dirichlet g.f.: zeta(s-1) * Product_{primes p} (1 + 1/(p^s - p - 1)). - Vaclav Kotesovec, Aug 22 2021

Extensions

Definition reedited (with formula) by Daniel Forgues, Nov 17 2009