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A060640 If n = Product p_i^e_i then a(n) = Product (1 + 2*p_i + 3*p_i^2 + ... + (e_i+1)*p_i^e_i).

%I A060640 #76 Jul 02 2025 16:02:01
%S A060640 1,5,7,17,11,35,15,49,34,55,23,119,27,75,77,129,35,170,39,187,105,115,
%T A060640 47,343,86,135,142,255,59,385,63,321,161,175,165,578,75,195,189,539,
%U A060640 83,525,87,391,374,235,95,903,162,430,245,459,107,710,253,735,273,295,119
%N A060640 If n = Product p_i^e_i then a(n) = Product (1 + 2*p_i + 3*p_i^2 + ... + (e_i+1)*p_i^e_i).
%C A060640 Equals row sums of triangle A143313. - _Gary W. Adamson_, Aug 06 2008
%C A060640 Equals row sums of triangle A127099. - _Gary W. Adamson_, Jul 27 2008
%C A060640 Sum of the divisors d2 from the ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - _Wesley Ivan Hurt_, Mar 22 2022
%D A060640 D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976, p. 120.
%H A060640 Seiichi Manyama, <a href="/A060640/b060640.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harry J. Smith)
%F A060640 a(n) = Sum_{d|n} d*tau(d), where tau(d) is the number of divisors of d, cf. A000005. a(n) = Sum_{d|n} d*sigma(n/d), where sigma(n)=sum of divisors of n, cf. A000203. - _Vladeta Jovovic_, Apr 23 2001
%F A060640 Multiplicative with a(p^e) = ((e+1)*p^{e+2} - (e+2)*p^{e+1} + 1) / (p-1)^2. Dirichlet g.f.: zeta(s)*zeta(s-1)^2. - _Franklin T. Adams-Watters_, Aug 03 2006
%F A060640 L.g.f.: Sum(A060640(n)*x^n/n) = -log( Product_{j>=1} P(x^j) ) where P(x) = Product_{k>=1} (1-x^k). - _Joerg Arndt_, May 03 2008
%F A060640 G.f.: Sum_{k>=1} k*tau(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Sep 06 2018
%F A060640 Sum_{k=1..n} a(k) ~ n^2/24 * ((4*gamma - 1)*Pi^2 + 2*Pi^2 * log(n) + 12*Zeta'(2)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Feb 01 2019
%e A060640 a(4) = a(2^2) = 1 + (2)*(2) + (3)*(2^2) = 17;
%e A060640 a(6) = a(2)*a(3) = (1 + (2)*(2))*(1+(2)*(3)) = (5)*(7) = 35.
%e A060640 a(6) = tau(1) + 2*tau(2) + 3*tau(3) + 6*tau(6) = 1 + 2*2 + 3*2 + 6*4 = 35.
%p A060640 A060640 := proc(n) local ans, i, j; ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*(1+sum((j+1)*ifactors(n)[2][i][1]^j,j=1..ifactors(n)[2][i][2])): od: RETURN(ans) end:
%t A060640 a[n_] := Total[#*DivisorSigma[1, n/#] & /@ Divisors[n]];
%t A060640 a /@ Range[59] (* _Jean-François Alcover_, May 19 2011, after _Vladeta Jovovic_ *)
%t A060640 f[p_, e_] := ((e + 1)*p^(e + 2) - (e + 2)*p^(e + 1) + 1)/(p - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Apr 10 2022 *)
%o A060640 (PARI) j=[]; for(n=1,200,j=concat(j,sumdiv(n,d,n/d*sigma(d)))); j
%o A060640 (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)/(1-p*X)^2)[n]) /* _Ralf Stephan_ */
%o A060640 (PARI) N=66; default(seriesprecision,N); x=z+O(z^(N+1))
%o A060640 c=sum(j=1,N,j*x^j); t=1/prod(j=1,N, eta(x^(j)));
%o A060640 t=log(t);t=serconvol(t,c);
%o A060640 Vec(t) /* _Joerg Arndt_, May 03 2008 */
%o A060640 (PARI) { for (n=1, 1000, write("b060640.txt", n, " ", direuler(p=2, n, 1/(1 - X)/(1 - p*X)^2)[n]); ) } /* _Harry J. Smith_, Jul 08 2009 */
%o A060640 (Haskell)
%o A060640 a060640 n = sum [d * a000005 d | d <- a027750_row n]
%o A060640 -- _Reinhard Zumkeller_, Feb 29 2012
%o A060640 (Sage)
%o A060640 def A060640(n) :
%o A060640     sigma = sloane.A000203
%o A060640     return add(sigma(k)*(n/k) for k in divisors(n))
%o A060640 [A060640(i) for i in (1..59)] # _Peter Luschny_, Sep 15 2012
%Y A060640 Cf. A000005, A000203, A001001, A006171, A038040 (Mobius transform), A049060, A057660, A057723, A327960 (Dirichlet inverse).
%Y A060640 Cf. also triangles A027750, A127099, A143313.
%K A060640 nonn,nice,easy,mult
%O A060640 1,2
%A A060640 _N. J. A. Sloane_, Apr 17 2001
%E A060640 More terms from _James Sellers_, _Vladeta Jovovic_ and _Matthew Conroy_, Apr 17 2001