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A003959 If n = Product p(k)^e(k) then a(n) = Product (p(k)+1)^e(k), a(1) = 1. 33

%I

%S 1,3,4,9,6,12,8,27,16,18,12,36,14,24,24,81,18,48,20,54,32,36,24,108,

%T 36,42,64,72,30,72,32,243,48,54,48,144,38,60,56,162,42,96,44,108,96,

%U 72,48,324,64,108,72,126,54,192,72,216,80,90,60,216,62,96,128,729,84,144,68

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

%C Completely multiplicative.

%C 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

%H T. D. Noe and Daniel Forgues, <a href="/A003959/b003959.txt">Table of n, a(n) for n = 1..100000</a> (first 1000 terms from T. D. Noe)

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%F Multiplicative with a(p^e) = (p+1)^e. - _David W. Wilson_, Aug 01, 2001.

%F sum(n>0, a(n)/n^s) = product(p prime, 1/(1-p^(-s)-p^(1-s)) ) (conjectured). - _Ralf Stephan_, Jul 07 2013

%p a:= n-> mul((i[1]+1)^i[2], i=ifactors(n)[2]):

%p seq(a(n), n=1..80); # _Alois P. Heinz_, Sep 13 2017

%t 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 *)

%o (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-X-p*X))[n]) /* _Ralf Stephan_ */

%o (Haskell)

%o a003959 1 = 1

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

%o -- _Reinhard Zumkeller_, Apr 09 2012

%Y Apart from initial terms, same as A064478.

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

%K nonn,easy,nice,mult

%O 1,2

%A _Marc LeBrun_

%E Definition reedited (with formula) by _Daniel Forgues_, Nov 17 2009

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Last modified June 20 09:12 EDT 2019. Contains 324234 sequences. (Running on oeis4.)