A003964 -id:A003964 - cp's OEIS frontend

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

1, 3, 6, 9, 15, 18, 28, 27, 36, 45, 66, 54, 91, 84, 90, 81, 153, 108, 190, 135, 168, 198, 276, 162, 225, 273, 216, 252, 435, 270, 496, 243, 396, 459, 420, 324, 703, 570, 546, 405, 861, 504, 946, 594, 540, 828, 1128, 486, 784, 675, 918, 819, 1431, 648, 990, 756

Offset: 1

Jonathan Vos Post, Oct 10 2007

There are analogs with the triangular numbers replaced by some other sequence, but this was chosen because of the parity coincidences of A034953.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A000217, A003958, A003959, A003960, A003961, A003964, A034953, A061142, A167338.

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 *)

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 */

for(n=1, 100, print1(direuler(p=2, n, 1 + (p^2 + p) / (2/X - p^2 - p))[n], ", ")) \\ Vaclav Kotesovec, Apr 05 2023

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).
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
a(n) = A167338(n)/A061142(n). - Vaclav Kotesovec, Jan 28 2025
Conjecture: Sum_{k=1..n} a(k) = O(n^3/log(n)). - Vaclav Kotesovec, Jan 28 2025

A357853 Fully multiplicative with a(prime(k)) = A000009(k+1).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 4, 2, 5, 3, 4, 1, 6, 4, 8, 2, 6, 4, 10, 2, 4, 5, 8, 3, 12, 4, 15, 1, 8, 6, 6, 4, 18, 8, 10, 2, 22, 6, 27, 4, 8, 10, 32, 2, 9, 4, 12, 5, 38, 8, 8, 3, 16, 12, 46, 4, 54, 15, 12, 1, 10, 8, 64, 6, 20, 6, 76, 4, 89, 18, 8, 8, 12, 10

Offset: 1

Gus Wiseman, Oct 28 2022

			We have 525 = prime(2)*prime(3)*prime(3)*prime(4) so a(525) = Q(3)*Q(4)*Q(4)*Q(5) = 2*2*2*3 = 24, where Q = A000009.

Other multiplicative sequences: A003961, A064988, A064989, A357852, A357980.
The non-strict version is A003964.
The unshifted horizontal version is A357978, non-strict A357977.
The unshifted version is A357982.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000009, A000720, A076610, A273873, A296150, A357979.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ptf[f_][n_]:=Product[f[i],{i,primeMS[n]}];
Array[ptf[PartitionsQ[#+1]&],100]

cp's OEIS Frontend

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

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