OFFSET
1,2
COMMENTS
There are analogs with the triangular numbers replaced by some other sequence, but this was chosen because of the parity coincidences of A034953.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
a((p_1)^e_1)*(p_2)^e_2)*...*(p_k)^e_k)) = (T((p_1))^e_1)*T((p_2))^e_2)*...*T((p_k))^e_k, where T(i) = A000217(i). a(p_i) = A034953(i).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 - 2/(p*(p+1))^(-1) = 2.12007865309570462566... . - Amiram Eldar, Dec 24 2022
Dirichlet g.f.: Product_{p prime} (1 + (p^2 + p) / (2*p^s - p^2 - p)). - Vaclav Kotesovec, Apr 05 2023
MATHEMATICA
f[p_, e_] := (p*(p + 1)/2)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Dec 24 2022 *)
PROG
(PARI) a(n)=my(f=factor(n)); prod(i=1, #f[, 1], binomial(f[i, 1]+1, 2)^f[i, 2]) /* Charles R Greathouse IV, Sep 09 2010 */
(PARI) for(n=1, 100, print1(direuler(p=2, n, 1 + (p^2 + p) / (2/X - p^2 - p))[n], ", ")) \\ Vaclav Kotesovec, Apr 05 2023
CROSSREFS
KEYWORD
mult,easy,nonn
AUTHOR
Jonathan Vos Post, Oct 10 2007
STATUS
approved