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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). 39
1, 5, 7, 17, 11, 35, 15, 49, 34, 55, 23, 119, 27, 75, 77, 129, 35, 170, 39, 187, 105, 115, 47, 343, 86, 135, 142, 255, 59, 385, 63, 321, 161, 175, 165, 578, 75, 195, 189, 539, 83, 525, 87, 391, 374, 235, 95, 903, 162, 430, 245, 459, 107, 710, 253, 735, 273, 295, 119 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Equals row sums of triangle A143313. - Gary W. Adamson, Aug 06 2008
Equals row sums of triangle A127099. - Gary W. Adamson, Jul 27 2008
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
REFERENCES
D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976, p. 120.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
FORMULA
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
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
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
G.f.: Sum_{k>=1} k*tau(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 06 2018
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
EXAMPLE
a(4) = a(2^2) = 1 + (2)*(2) + (3)*(2^2) = 17;
a(6) = a(2)*a(3) = (1 + (2)*(2))*(1+(2)*(3)) = (5)*(7) = 35.
a(6) = tau(1) + 2*tau(2) + 3*tau(3) + 6*tau(6) = 1 + 2*2 + 3*2 + 6*4 = 35.
MAPLE
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:
MATHEMATICA
a[n_] := Total[#*DivisorSigma[1, n/#] & /@ Divisors[n]];
a /@ Range[59] (* Jean-François Alcover, May 19 2011, after Vladeta Jovovic *)
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 *)
PROG
(PARI) j=[]; for(n=1, 200, j=concat(j, sumdiv(n, d, n/d*sigma(d)))); j
(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X)/(1-p*X)^2)[n]) /* Ralf Stephan */
(PARI) N=66; default(seriesprecision, N); x=z+O(z^(N+1))
c=sum(j=1, N, j*x^j); t=1/prod(j=1, N, eta(x^(j)));
t=log(t); t=serconvol(t, c);
Vec(t) /* Joerg Arndt, May 03 2008 */
(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 */
(Haskell)
a060640 n = sum [d * a000005 d | d <- a027750_row n]
-- Reinhard Zumkeller, Feb 29 2012
(Sage)
def A060640(n) :
sigma = sloane.A000203
return add(sigma(k)*(n/k) for k in divisors(n))
[A060640(i) for i in (1..59)] # Peter Luschny, Sep 15 2012
CROSSREFS
Cf. A143313. - Gary W. Adamson, Aug 06 2008
Cf. A127099.
Cf. A027750.
Sequence in context: A186710 A276717 A318491 * A064944 A341038 A070372
KEYWORD
nonn,nice,easy,mult
AUTHOR
N. J. A. Sloane, Apr 17 2001
EXTENSIONS
More terms from James A. Sellers, Vladeta Jovovic and Matthew Conroy, Apr 17 2001
STATUS
approved

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Last modified February 21 09:14 EST 2024. Contains 370228 sequences. (Running on oeis4.)