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A003959 If n = Product p(k)^e(k) then a(n) = Product (p(k)+1)^e(k), a(1) = 1. 32
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

T. D. Noe and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)

Index to divisibility sequences

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

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-Fran├žois Alcover, Apr 22 2011 *)

PROG

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

(Haskell)

a003959 1 = 1

a003959 n = product $ map (+ 1) $ a027746_row n

-- Reinhard Zumkeller, Apr 09 2012

CROSSREFS

Apart from initial terms, same as A064478.

Cf. A003958, A063441, A168065, A168066, A163407, A027746.

Sequence in context: A249187 A084425 A168512 * A168341 A083111 A132065

Adjacent sequences:  A003956 A003957 A003958 * A003960 A003961 A003962

KEYWORD

nonn,easy,nice,mult

AUTHOR

Marc LeBrun

EXTENSIONS

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

STATUS

approved

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Last modified May 26 19:44 EDT 2019. Contains 323597 sequences. (Running on oeis4.)