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%I #71 Oct 29 2023 06:37:35
%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 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
%F 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
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = A065488/2 = 1/(2*A005596) = 1.3370563627850107544802059152227440187511993141988459926... - _Vaclav Kotesovec_, Jul 17 2021
%F From _Thomas Scheuerle_, Jul 19 2021: (Start)
%F a(n) = gcd(A166642(n), A166643(n)).
%F a(n) = A166642(n)/A061142(n).
%F a(n) = A166643(n)/A165824(n).
%F a(n) = A166644(n)/A165825(n).
%F a(n) = A166645(n)/A165826(n).
%F a(n) = A166646(n)/A165827(n).
%F a(n) = A166647(n)/A165828(n).
%F a(n) = A166649(n)/A165830(n).
%F a(n) = A166650(n)/A165831(n).
%F a(n) = A167351(n)/A166590(n). (End)
%F Dirichlet g.f.: zeta(s-1) * Product_{primes p} (1 + 1/(p^s - p - 1)). - _Vaclav Kotesovec_, Aug 22 2021
%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-François 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, A027746, A036987, A061142, A163407.
%Y Cf. A063441, A165824, A165825, A165826, A165827.
%Y Cf. A165828, A165829, A165831, A167350, A166590.
%Y Cf. A166642, A166643, A166644, A166645, A166646.
%Y Cf. A166647, A166649, A166650, A166586, A165830.
%Y Cf. A167351, A168065, A168066.
%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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