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A167338
Totally multiplicative sequence with a(p) = p*(p+1) = p^2+p for prime p.
2
1, 6, 12, 36, 30, 72, 56, 216, 144, 180, 132, 432, 182, 336, 360, 1296, 306, 864, 380, 1080, 672, 792, 552, 2592, 900, 1092, 1728, 2016, 870, 2160, 992, 7776, 1584, 1836, 1680, 5184, 1406, 2280, 2184, 6480, 1722, 4032, 1892, 4752, 4320, 3312, 2256, 15552
OFFSET
1,2
LINKS
FORMULA
Multiplicative with a(p^e) = (p*(p+1))^e.
If n = Product p(k)^e(k) then a(n) = Product (p(k)*(p(k)+1))^e(k).
a(n) = n * A003959(n).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + p - 1)) = A065489 = 1.419562880505485919317235861789735359166071586305122542698983695564330971... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 - 2/p^2 - 1/p^3) = 0.8913709085... . - Amiram Eldar, Dec 15 2022, c = A065488/3. - Vaclav Kotesovec, Apr 05 2023
Dirichlet g.f.: zeta(s-2) * Product_{p prime} (1 + 1/(p^(s-1) - p - 1)). - Vaclav Kotesovec, Apr 05 2023
MATHEMATICA
a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 1)^fi[[All, 2]])); Table[a[n]*n, {n, 1, 100}] (* G. C. Greubel, Jun 06 2016 *)
PROG
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 1/(1/X/p - p - 1))/(1 - p^2*X))[n], ", ")) \\ Vaclav Kotesovec, Apr 05 2023
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Jaroslav Krizek, Nov 01 2009
STATUS
approved