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A133205 Fully multiplicative with a(p) = p*(p+1)/2 for prime p.

%I A133205 #29 Jan 28 2025 08:06:08
%S A133205 1,3,6,9,15,18,28,27,36,45,66,54,91,84,90,81,153,108,190,135,168,198,
%T A133205 276,162,225,273,216,252,435,270,496,243,396,459,420,324,703,570,546,
%U A133205 405,861,504,946,594,540,828,1128,486,784,675,918,819,1431,648,990,756
%N A133205 Fully multiplicative with a(p) = p*(p+1)/2 for prime p.
%C A133205 There are analogs with the triangular numbers replaced by some other sequence, but this was chosen because of the parity coincidences of A034953.
%H A133205 Amiram Eldar, <a href="/A133205/b133205.txt">Table of n, a(n) for n = 1..10000</a>
%F A133205 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(prime(i)) = A034953(i).
%F A133205 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 - 2/(p*(p+1)))^(-1) = 2.12007865309570462566... . - _Amiram Eldar_, Dec 24 2022
%F A133205 Dirichlet g.f.: Product_{p prime} (1 + (p^2 + p) / (2*p^s - p^2 - p)). - _Vaclav Kotesovec_, Apr 05 2023
%F A133205 a(n) = A167338(n)/A061142(n). - _Vaclav Kotesovec_, Jan 28 2025
%F A133205 Conjecture: Sum_{k=1..n} a(k) = O(n^3/log(n)). - _Vaclav Kotesovec_, Jan 28 2025
%t A133205 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 *)
%o A133205 (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 */
%o A133205 (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
%Y A133205 Cf. A000040, A000217, A003958, A003959, A003960, A003961, A003964, A034953, A061142, A167338.
%K A133205 mult,easy,nonn
%O A133205 1,2
%A A133205 _Jonathan Vos Post_, Oct 10 2007