login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation to keep the OEIS running. In 2018 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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). 25
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]

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

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-Fran├žois Alcover, May 19 2011, after Vladeta Jovovic *)

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. A000005, A000203, A001001, A006171, A038040, A049060, A057660, A057723.

Cf. A143313. - Gary W. Adamson, Aug 06 2008

Cf. A127099.

Cf. A027750.

Sequence in context: A186710 A276717 A318491 * A064944 A070372 A082818

Adjacent sequences:  A060637 A060638 A060639 * A060641 A060642 A060643

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 12 19:33 EST 2018. Contains 318081 sequences. (Running on oeis4.)